# How to smooth the graph $\chi(t)=\sen(\pi t)\exp\left(\dfrac{-1}{\sen^2(\pi t)}\right)$

I am graphing these two functions $$\chi(t)=\sin(\pi t)\exp\left(\dfrac{-1}{\sin^2(\pi t)}\right)$$ and $\rho(t)=\sqrt{\frac{1}{A}\int_0^{t+1}\chi(x)dx}$, where $A=\int_0^1 \chi(x)dx$.

I only succeeded in graphing the first one but I don't get anything smooth, I know that the function $\chi$ is not defined in the integers but it can be fixed by putting $\chi(x)=0$ when x is integer (I don't know how to graph all the function).

\documentclass[border=5mm]{standalone}
\usepackage{tikz}
\begin{document}

\begin{tikzpicture}
\draw[-stealth] (-2.5,0) -- (2.5,0) node[below]{$t$};;
\foreach \X in {-2,-1,1,2} {\draw (\X,0.1) -- (\X,-0.1) node[below]{$\X$}; };
\draw[-stealth] (0,-1) -- (0,1) node[left]{$\chi(t)$};

\draw[smooth,domain = -2:2, color=blue] plot(\x,{exp(-1/(sin(deg(\x*pi)))^2)*sin(deg(\x*pi))});
\end{tikzpicture}
\end{document}


Plot:

As you can see in my code I tried to graph it and the error arises from the points of discontinuity that I mentioned above. I would like it to be graphed without errors and also to be smoother, for example look compared to geogebra.

Regarding the second function I have no idea how to graph an integral, I saw some related questions but when compiling I did not get any results. I am sorry if there are errors, I am something new graphing in LaTeX.

Update: I have obtained the $chi(t)$ function, but the code is very long, and for the sake of not distorting the original question, I will put my solution in an answer. Bearing in mind that even though I don't get the $\rpho(t)$ graph, I'm sorry if what I'm doing is wrong.

• If you want accurate/smooth graphs, calculate them using another language (C++ or MATLAB) and store the numbers in a comma delimited table. Commented Nov 9, 2021 at 15:29
• There are a number of ways to calculate integrals which do not have closed form solutions. My favorite is to approximate the function with a polynomial and integrate the polynomial. Commented Nov 9, 2021 at 15:32
• @JohnKormylo Approximating may be a harder math problem. I think Asymptote with avaiable simpson(f,a,b) function (for integral from  a to b of f) is a good way Commented Nov 9, 2021 at 18:50
• @BlackMild I also thought about it but the problem is that my upper integration limit is variable, how do I do it in that case? Commented Nov 9, 2021 at 18:52

A try with Asymptote.

Compile on http://asymptote.ualberta.ca/.

Note that test=64 is a lucky number for my computer.

import graph;
size(300,200,false);

real chi(real t)
{
real a=sin(pi*t);
return (a != 0) ? a*exp(-1/a^2) : 0;
}
real rho(real t)
{
return sqrt(simpson(chi,0,t+1)/simpson(chi,0,1));
}

int test=64;
guide g1=graph(chi,0,3.5,300,Spline);
guide g2=graph(rho,0,3.5,test,Spline);
draw(Label("$\chi(t)$",Relative(.4),black),g1,red+0.7bp);
draw(Label("$\rho(t)$",Relative(.12),LeftSide,black),g2,blue+1bp);

xaxis(Label("$x$",position=EndPoint, align=NE), xmin=-0.3, xmax=3.5,
Ticks(scale(.85)*Label(), NoZero, endlabel=false, Step=0.5,
end=false, Size=.8mm, pTick=black),
Arrow);
yaxis(Label("$y$",position=EndPoint, align=NE), ymin=-0.4, ymax=1.2,
Ticks(scale(.85)*Label(), NoZero, endlabel=false, Step=0.2,
end=false, Size=.8mm, pTick=black),
Arrow);


• Thank you very much, great solution Commented Nov 9, 2021 at 19:10
• definition of chi should be (a != 0) ? a*exp(-1/a^2) : 0; instead of (t != 0) ? a*exp(-1/a^2) : 0;. Another way is real chi(real t){ if (t-floor(t)==0) return 0; else return sin(pi*t)*exp(-1/(sin(pi*t))^2); } Commented Nov 9, 2021 at 19:25
• @BlackMild Yes, edited. Commented Nov 9, 2021 at 20:34

This I have obtained for function $chi(t)$:

\documentclass[border=1mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.15}
\usepackage{mathrsfs}
\usetikzlibrary{arrows}
\pagestyle{empty}
\begin{document}
\definecolor{qqqqff}{rgb}{0,0,1}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1cm,y=1cm]
\begin{axis}[
x=1cm,y=2cm,
axis lines=middle,
xmin=-2.592785508994096,
xmax=2.548506587329168,
ymin=-0.6,
ymax=0.6,
xtick={0,1,...,0},
ytick={0,0.5,...,0},]
\clip(-2.592785508994096,-0.4025540845460146) rectangle (2.548506587329168,0.4257312538518619);
\draw[color=qqqqff] (-2.001536917916929,0) -- (-1.9886836876761207,0);
\draw[color=qqqqff] (-1.9886836876761207,0) -- (-1.9758304574353125,0);
\draw[color=qqqqff] (-1.9758304574353125,0) -- (-1.9629772271945043,0);
\draw[color=qqqqff] (-1.9629772271945043,0) -- (-1.950123996953696,0);
\draw[color=qqqqff] (-1.950123996953696,0) -- (-1.937270766712888,0);
\draw[color=qqqqff] (-1.937270766712888,0) -- (-1.9244175364720797,0);
\draw[color=qqqqff] (-1.9244175364720797,0) -- (-1.9115643062312715,0);
\draw[color=qqqqff] (-1.9115643062312715,0) -- (-1.8987110759904633,0);
\draw[color=qqqqff] (-1.8987110759904633,0) -- (-1.885857845749655,0);
\draw[color=qqqqff] (-1.885857845749655,0) -- (-1.873004615508847,0);
\draw[color=qqqqff] (-1.873004615508847,0) -- (-1.8601513852680387,0.0016915696259532037);
\draw[color=qqqqff] (-1.8601513852680387,0.0016915696259532037) -- (-1.8472981550272305,0.004221067810271749);
\draw[color=qqqqff] (-1.8472981550272305,0.004221067810271749) -- (-1.8344449247864223,0.008668137368654873);
\draw[color=qqqqff] (-1.8344449247864223,0.008668137368654873) -- (-1.821591694545614,0.015444822624142986);
\draw[color=qqqqff] (-1.821591694545614,0.015444822624142986) -- (-1.808738464304806,0.024749960583867154);
\draw[color=qqqqff] (-1.808738464304806,0.024749960583867154) -- (-1.7958852340639977,0.03657404119818469);
\draw[color=qqqqff] (-1.7958852340639977,0.03657404119818469) -- (-1.7830320038231895,0.050740697110255015);
\draw[color=qqqqff] (-1.7830320038231895,0.050740697110255015) -- (-1.7701787735823813,0.06695938106778791);
\draw[color=qqqqff] (-1.7701787735823813,0.06695938106778791) -- (-1.757325543341573,0.08487406232481962);
\draw[color=qqqqff] (-1.757325543341573,0.08487406232481962) -- (-1.744472313100765,0.10410163055763474);
\draw[color=qqqqff] (-1.744472313100765,0.10410163055763474) -- (-1.7316190828599567,0.12425904918259241);
\draw[color=qqqqff] (-1.7316190828599567,0.12425904918259241) -- (-1.7187658526191485,0.14498078382607177);
\draw[color=qqqqff] (-1.7187658526191485,0.14498078382607177) -- (-1.7059126223783403,0.16592878720703774);
\draw[color=qqqqff] (-1.7059126223783403,0.16592878720703774) -- (-1.693059392137532,0.18679723362822878);
\draw[color=qqqqff] (-1.693059392137532,0.18679723362822878) -- (-1.680206161896724,0.20731379187716573);
\draw[color=qqqqff] (-1.680206161896724,0.20731379187716573) -- (-1.6673529316559157,0.2272387688006349);
\draw[color=qqqqff] (-1.6673529316559157,0.2272387688006349) -- (-1.6544997014151075,0.24636305667353747);
\draw[color=qqqqff] (-1.6544997014151075,0.24636305667353747) -- (-1.6416464711742993,0.26450550667944267);
\draw[color=qqqqff] (-1.6416464711742993,0.26450550667944267) -- (-1.628793240933491,0.2815101247897043);
\draw[color=qqqqff] (-1.628793240933491,0.2815101247897043) -- (-1.615940010692683,0.2972433295678396);
\draw[color=qqqqff] (-1.615940010692683,0.2972433295678396) -- (-1.6030867804518747,0.31159140677348685);
\draw[color=qqqqff] (-1.6030867804518747,0.31159140677348685) -- (-1.5902335502110665,0.32445822823067255);
\draw[color=qqqqff] (-1.5902335502110665,0.32445822823067255) -- (-1.5773803199702583,0.3357632606995007);
\draw[color=qqqqff] (-1.5773803199702583,0.3357632606995007) -- (-1.56452708972945,0.3454398659933108);
\draw[color=qqqqff] (-1.56452708972945,0.3454398659933108) -- (-1.551673859488642,0.35343388037100765);
\draw[color=qqqqff] (-1.551673859488642,0.35343388037100765) -- (-1.5388206292478337,0.35970245524844974);
\draw[color=qqqqff] (-1.5388206292478337,0.35970245524844974) -- (-1.5259673990070255,0.3642131398076329);
\draw[color=qqqqff] (-1.5259673990070255,0.3642131398076329) -- (-1.5131141687662173,0.36694318737287357);
\draw[color=qqqqff] (-1.5131141687662173,0.36694318737287357) -- (-1.500260938525409,0.3678790703431535);
\draw[color=qqqqff] (-1.500260938525409,0.3678790703431535) -- (-1.487407708284601,0.367016192317628);
\draw[color=qqqqff] (-1.487407708284601,0.367016192317628) -- (-1.4745544780437927,0.3643587903995568);
\draw[color=qqqqff] (-1.4745544780437927,0.3643587903995568) -- (-1.4617012478029845,0.3599200252532819);
\draw[color=qqqqff] (-1.4617012478029845,0.3599200252532819) -- (-1.4488480175621763,0.3537222611507955);
\draw[color=qqqqff] (-1.4488480175621763,0.3537222611507955) -- (-1.435994787321368,0.3457975428396393);
\draw[color=qqqqff] (-1.435994787321368,0.3457975428396393) -- (-1.42314155708056,0.3361882804261461);
\draw[color=qqqqff] (-1.42314155708056,0.3361882804261461) -- (-1.4102883268397517,0.3249481573430678);
\draw[color=qqqqff] (-1.4102883268397517,0.3249481573430678) -- (-1.3974350965989435,0.3121432794396657);
\draw[color=qqqqff] (-1.3974350965989435,0.3121432794396657) -- (-1.3845818663581353,0.2978535846074099);
\draw[color=qqqqff] (-1.3845818663581353,0.2978535846074099) -- (-1.371728636117327,0.2821745310303907);
\draw[color=qqqqff] (-1.371728636117327,0.2821745310303907) -- (-1.358875405876519,0.2652190763923943);
\draw[color=qqqqff] (-1.358875405876519,0.2652190763923943) -- (-1.3460221756357107,0.247119947522008);
\draw[color=qqqqff] (-1.3460221756357107,0.247119947522008) -- (-1.3331689453949025,0.22803217601925163);
\draw[color=qqqqff] (-1.3331689453949025,0.22803217601925163) -- (-1.3203157151540943,0.20813583452673357);
\draw[color=qqqqff] (-1.3203157151540943,0.20813583452673357) -- (-1.307462484913286,0.1876388421441786);
\draw[color=qqqqff] (-1.307462484913286,0.1876388421441786) -- (-1.294609254672478,0.16677960461303723);
\draw[color=qqqqff] (-1.294609254672478,0.16677960461303723) -- (-1.2817560244316697,0.14582910057967377);
\draw[color=qqqqff] (-1.2817560244316697,0.14582910057967377) -- (-1.2689027941908615,0.12509180236327802);
\draw[color=qqqqff] (-1.2689027941908615,0.12509180236327802) -- (-1.2560495639500533,0.10490451249611192);
\draw[color=qqqqff] (-1.2560495639500533,0.10490451249611192) -- (-1.243196333709245,0.08563180148775335);
\draw[color=qqqqff] (-1.243196333709245,0.08563180148775335) -- (-1.230343103468437,0.06765627665778033);
\draw[color=qqqqff] (-1.230343103468437,0.06765627665778033) -- (-1.2174898732276287,0.0513615015926554);
\draw[color=qqqqff] (-1.2174898732276287,0.0513615015926554) -- (-1.2046366429868205,0.03710527677724);
\draw[color=qqqqff] (-1.2046366429868205,0.03710527677724) -- (-1.1917834127460123,0.02518169852620193);
\draw[color=qqqqff] (-1.1917834127460123,0.02518169852620193) -- (-1.178930182505204,0.015772805357213045);
\draw[color=qqqqff] (-1.178930182505204,0.015772805357213045) -- (-1.166076952264396,0.008895837238822916);
\draw[color=qqqqff] (-1.166076952264396,0.008895837238822916) -- (-1.1532237220235877,0.004360859138376867);
\draw[color=qqqqff] (-1.1532237220235877,0.004360859138376867) -- (-1.1403704917827795,0.001763826698339104);
\draw[color=qqqqff] (-1.1403704917827795,0.001763826698339104) -- (-1.1275172615419713,0);
\draw[color=qqqqff] (-1.1275172615419713,0) -- (-1.114664031301163,0);
\draw[color=qqqqff] (-1.114664031301163,0) -- (-1.101810801060355,0);
\draw[color=qqqqff] (-1.101810801060355,0) -- (-1.0889575708195467,0);
\draw[color=qqqqff] (-1.0889575708195467,0) -- (-1.0761043405787385,0);
\draw[color=qqqqff] (-1.0761043405787385,0) -- (-1.0632511103379303,0);
\draw[color=qqqqff] (-1.0632511103379303,0) -- (-1.050397880097122,0);
\draw[color=qqqqff] (-1.050397880097122,0) -- (-1.037544649856314,0);
\draw[color=qqqqff] (-1.037544649856314,0) -- (-1.0246914196155057,0);
\draw[color=qqqqff] (-1.0246914196155057,0) -- (-1.0118381893746975,0);
\draw[color=qqqqff] (-1.0118381893746975,0) -- (-0.9989849591338893,0);
\draw[color=qqqqff] (-0.9989849591338893,0) -- (-0.9861317288930811,0);
\draw[color=qqqqff] (-0.9861317288930811,0) -- (-0.9732784986522729,0);
\draw[color=qqqqff] (-0.9732784986522729,0) -- (-0.9604252684114647,0);
\draw[color=qqqqff] (-0.9604252684114647,0) -- (-0.9475720381706565,0);
\draw[color=qqqqff] (-0.9475720381706565,0) -- (-0.9347188079298483,0);
\draw[color=qqqqff] (-0.9347188079298483,0) -- (-0.9218655776890401,0);
\draw[color=qqqqff] (-0.9218655776890401,0) -- (-0.9090123474482319,0);
\draw[color=qqqqff] (-0.9090123474482319,0) -- (-0.8961591172074237,0);
\draw[color=qqqqff] (-0.8961591172074237,0) -- (-0.8833058869666155,0);
\draw[color=qqqqff] (-0.8833058869666155,0) -- (-0.8704526567258073,0);
\draw[color=qqqqff] (-0.8704526567258073,0) -- (-0.8575994264849991,-0.0020669120110518094);
\draw[color=qqqqff] (-0.8575994264849991,-0.0020669120110518094) -- (-0.8447461962441909,-0.004935553593396952);
\draw[color=qqqqff] (-0.8447461962441909,-0.004935553593396952) -- (-0.8318929660033827,-0.009818772744365982);
\draw[color=qqqqff] (-0.8318929660033827,-0.009818772744365982) -- (-0.8190397357625745,-0.017088664180427565);
\draw[color=qqqqff] (-0.8190397357625745,-0.017088664180427565) -- (-0.8061865055217663,-0.026900714976924547);
\draw[color=qqqqff] (-0.8061865055217663,-0.026900714976924547) -- (-0.7933332752809581,-0.0392083260151868);
\draw[color=qqqqff] (-0.7933332752809581,-0.0392083260151868) -- (-0.7804800450401499,-0.053808243802419574);
\draw[color=qqqqff] (-0.7804800450401499,-0.053808243802419574) -- (-0.7676268147993417,-0.07039332592951228);
\draw[color=qqqqff] (-0.7676268147993417,-0.07039332592951228) -- (-0.7547735845585335,-0.0885994759716792);
\draw[color=qqqqff] (-0.7547735845585335,-0.0885994759716792) -- (-0.7419203543177253,-0.10804177721651512);
\draw[color=qqqqff] (-0.7419203543177253,-0.10804177721651512) -- (-0.7290671240769171,-0.12833955168970812);
\draw[color=qqqqff] (-0.7290671240769171,-0.12833955168970812) -- (-0.7162138938361089,-0.14913212423379527);
\draw[color=qqqqff] (-0.7162138938361089,-0.14913212423379527) -- (-0.7033606635953007,-0.17008760104962245);
\draw[color=qqqqff] (-0.7033606635953007,-0.17008760104962245) -- (-0.6905074333544925,-0.1909067893639711);
\draw[color=qqqqff] (-0.6905074333544925,-0.1909067893639711) -- (-0.6776542031136843,-0.2113239554205922);
\draw[color=qqqqff] (-0.6776542031136843,-0.2113239554205922) -- (-0.6648009728728761,-0.231105667493171);
\draw[color=qqqqff] (-0.6648009728728761,-0.231105667493171) -- (-0.6519477426320679,-0.25004858811397707);
\draw[color=qqqqff] (-0.6519477426320679,-0.25004858811397707) -- (-0.6390945123912597,-0.26797678662596797);
\draw[color=qqqqff] (-0.6390945123912597,-0.26797678662596797) -- (-0.6262412821504515,-0.2847389322691392);
\draw[color=qqqqff] (-0.6262412821504515,-0.2847389322691392) -- (-0.6133880519096433,-0.30020558292094385);
\draw[color=qqqqff] (-0.6133880519096433,-0.30020558292094385) -- (-0.6005348216688351,-0.3142666884335657);
\draw[color=qqqqff] (-0.6005348216688351,-0.3142666884335657) -- (-0.5876815914280269,-0.32682936601475254);
\draw[color=qqqqff] (-0.5876815914280269,-0.32682936601475254) -- (-0.5748283611872187,-0.3378159673775671);
\draw[color=qqqqff] (-0.5748283611872187,-0.3378159673775671) -- (-0.5619751309464105,-0.3471624355322722);
\draw[color=qqqqff] (-0.5619751309464105,-0.3471624355322722) -- (-0.5491219007056023,-0.3548169375890639);
\draw[color=qqqqff] (-0.5491219007056023,-0.3548169375890639) -- (-0.5362686704647941,-0.3607387550323308);
\draw[color=qqqqff] (-0.5362686704647941,-0.3607387550323308) -- (-0.5234154402239859,-0.36489741212552446);
\draw[color=qqqqff] (-0.5234154402239859,-0.36489741212552446) -- (-0.5105622099831777,-0.367272024793215);
\draw[color=qqqqff] (-0.5105622099831777,-0.367272024793215) -- (-0.49770897974236955,-0.3678508554763816);
\draw[color=qqqqff] (-0.49770897974236955,-0.3678508554763816) -- (-0.4848557495015614,-0.366631063431549);
\draw[color=qqqqff] (-0.4848557495015614,-0.366631063431549) -- (-0.47200251926075326,-0.3636186443572912);
\draw[color=qqqqff] (-0.47200251926075326,-0.3636186443572912) -- (-0.4591492890199451,-0.3588285578470439);
\draw[color=qqqqff] (-0.4591492890199451,-0.3588285578470439) -- (-0.446296058779137,-0.35228504582698056);
\draw[color=qqqqff] (-0.446296058779137,-0.35228504582698056) -- (-0.43344282853832883,-0.3440221497017378);
\draw[color=qqqqff] (-0.43344282853832883,-0.3440221497017378) -- (-0.4205895982975207,-0.33408443822053846);
\draw[color=qqqqff] (-0.4205895982975207,-0.33408443822053846) -- (-0.40773636805671254,-0.32252796181156607);
\draw[color=qqqqff] (-0.40773636805671254,-0.32252796181156607) -- (-0.3948831378159044,-0.3094214518505393);
\draw[color=qqqqff] (-0.3948831378159044,-0.3094214518505393) -- (-0.38202990757509625,-0.29484778427302866);
\draw[color=qqqqff] (-0.38202990757509625,-0.29484778427302866) -- (-0.3691766773342881,-0.27890572489835463);
\draw[color=qqqqff] (-0.3691766773342881,-0.27890572489835463) -- (-0.35632344709347996,-0.2617119669148593);
\draw[color=qqqqff] (-0.35632344709347996,-0.2617119669148593) -- (-0.3434702168526718,-0.24340345629301932);
\draw[color=qqqqff] (-0.3434702168526718,-0.24340345629301932) -- (-0.3306169866118637,-0.2241399741266862);
\draw[color=qqqqff] (-0.3306169866118637,-0.2241399741266862) -- (-0.31776375637105553,-0.2041068997562384);
\draw[color=qqqqff] (-0.31776375637105553,-0.2041068997562384) -- (-0.3049105261302474,-0.18351800609017616);
\draw[color=qqqqff] (-0.3049105261302474,-0.18351800609017616) -- (-0.29205729588943924,-0.1626180267281583);
\draw[color=qqqqff] (-0.29205729588943924,-0.1626180267281583) -- (-0.2792040656486311,-0.1416845679547785);
\draw[color=qqqqff] (-0.2792040656486311,-0.1416845679547785) -- (-0.26635083540782295,-0.12102870014445041);
\draw[color=qqqqff] (-0.26635083540782295,-0.12102870014445041) -- (-0.2534976051670148,-0.10099323806373346);
\draw[color=qqqqff] (-0.2534976051670148,-0.10099323806373346) -- (-0.24064437492620663,-0.08194730793490931);
\draw[color=qqqqff] (-0.24064437492620663,-0.08194730793490931) -- (-0.22779114468539846,-0.064275340654999);
\draw[color=qqqqff] (-0.22779114468539846,-0.064275340654999) -- (-0.2149379144445903,-0.04835825321862172);
\draw[color=qqqqff] (-0.2149379144445903,-0.04835825321862172) -- (-0.20208468420378212,-0.034544582106425976);
\draw[color=qqqqff] (-0.20208468420378212,-0.034544582106425976) -- (-0.18923145396297394,-0.023110292435982397);
\draw[color=qqqqff] (-0.18923145396297394,-0.023110292435982397) -- (-0.17637822372216577,-0.014208846608098959);
\draw[color=qqqqff] (-0.17637822372216577,-0.014208846608098959) -- (-0.1635249934813576,-0.007819014708058283);
\draw[color=qqqqff] (-0.1635249934813576,-0.007819014708058283) -- (-0.15067176324054943,-0.00370724654575138);
\draw[color=qqqqff] (-0.15067176324054943,-0.00370724654575138) -- (-0.13781853299974126,-0.0014313536491768481);
\draw[color=qqqqff] (-0.13781853299974126,-0.0014313536491768481) -- (-0.1249653027589331,0);
\draw[color=qqqqff] (-0.1249653027589331,0) -- (-0.11211207251812494,0);
\draw[color=qqqqff] (-0.11211207251812494,0) -- (-0.09925884227731678,0);
\draw[color=qqqqff] (-0.09925884227731678,0) -- (-0.08640561203650862,0);
\draw[color=qqqqff] (-0.08640561203650862,0) -- (-0.07355238179570046,0);
\draw[color=qqqqff] (-0.07355238179570046,0) -- (-0.060699151554892306,0);
\draw[color=qqqqff] (-0.060699151554892306,0) -- (-0.04784592131408415,0);
\draw[color=qqqqff] (-0.04784592131408415,0) -- (-0.03499269107327599,0);
\draw[color=qqqqff] (-0.03499269107327599,0) -- (-0.02213946083246783,0);
\draw[color=qqqqff] (-0.02213946083246783,0) -- (-0.00928623059165967,0);
\draw[color=qqqqff] (-0.00928623059165967,0) -- (0.0035669996491484897,0);
\draw[color=qqqqff] (0.0035669996491484897,0) -- (0.01642022988995665,0);
\draw[color=qqqqff] (0.01642022988995665,0) -- (0.02927346013076481,0);
\draw[color=qqqqff] (0.02927346013076481,0) -- (0.04212669037157297,0);
\draw[color=qqqqff] (0.04212669037157297,0) -- (0.05497992061238113,0);
\draw[color=qqqqff] (0.05497992061238113,0) -- (0.06783315085318929,0);
\draw[color=qqqqff]  (0.06783315085318929,0) -- (0.08068638109399745,0);
\draw[color=qqqqff]  (0.08068638109399745,0) -- (0.09353961133480561,0);
\draw[color=qqqqff]  (0.09353961133480561,0) -- (0.10639284157561377,0);
\draw[color=qqqqff]  (0.10639284157561377,0) -- (0.11924607181642193,0);
\draw[color=qqqqff]  (0.11924607181642193,0) -- (0.13209930205723008,0);
\draw[color=qqqqff]  (0.13209930205723008,0) -- (0.14495253229803826,0.002500441204079152);
\draw[color=qqqqff]  (0.14495253229803826,0.002500441204079152) -- (0.15780576253884643,0.00573002939586416);
\draw[color=qqqqff]  (0.15780576253884643,0.00573002939586416) -- (0.1706589927796546,0.01106420133564719);
\draw[color=qqqqff]  (0.1706589927796546,0.01106420133564719) -- (0.18351222302046277,0.018833369073369927);
\draw[color=qqqqff]  (0.18351222302046277,0.018833369073369927) -- (0.19636545326127094,0.029150342359197095);
\draw[color=qqqqff]  (0.19636545326127094,0.029150342359197095) -- (0.20921868350207912,0.041933299681958726);
\draw[color=qqqqff]  (0.20921868350207912,0.041933299681958726) -- (0.2220719137428873,0.056954221697264995);
\draw[color=qqqqff]  (0.2220719137428873,0.056954221697264995) -- (0.23492514398369546,0.07389123515163279);
\draw[color=qqqqff]  (0.23492514398369546,0.07389123515163279) -- (0.24777837422450363,0.0923736021117788);
\draw[color=qqqqff]  (0.24777837422450363,0.0923736021117788) -- (0.2606316044653118,0.11201558109248587);
\draw[color=qqqqff]  (0.2606316044653118,0.11201558109248587) -- (0.27348483470611995,0.1324394665974412);
\draw[color=qqqqff]  (0.27348483470611995,0.1324394665974412) -- (0.2863380649469281,0.1532897819425046);
\draw[color=qqqqff]  (0.2863380649469281,0.1532897819425046) -- (0.29919129518773624,0.17424093597016993);
\draw[color=qqqqff]  (0.29919129518773624,0.17424093597016993) -- (0.3120445254285444,0.1950003948101426);
\draw[color=qqqqff]  (0.3120445254285444,0.1950003948101426) -- (0.32489775566935253,0.21530897386526573);
\draw[color=qqqqff]  (0.32489775566935253,0.21530897386526573) -- (0.3377509859101607,0.23493941403712268);
\draw[color=qqqqff]  (0.3377509859101607,0.23493941403712268) -- (0.3506042161509688,0.25369404106538895);
\draw[color=qqqqff]  (0.3506042161509688,0.25369404106538895) -- (0.36345744639177696,0.27140203114366884);
\draw[color=qqqqff]  (0.36345744639177696,0.27140203114366884) -- (0.3763106766325851,0.28791660954012716);
\draw[color=qqqqff]  (0.3763106766325851,0.28791660954012716) -- (0.38916390687339325,0.3031123748547486);
\draw[color=qqqqff]  (0.38916390687339325,0.3031123748547486) -- (0.4020171371142014,0.31688285328074733);
\draw[color=qqqqff]  (0.4020171371142014,0.31688285328074733) -- (0.41487036735500954,0.3291383312264683);
\draw[color=qqqqff]  (0.41487036735500954,0.3291383312264683) -- (0.4277235975958177,0.33980398062743544);
\draw[color=qqqqff]  (0.4277235975958177,0.33980398062743544) -- (0.44057682783662583,0.3488182718553158);
\draw[color=qqqqff]  (0.44057682783662583,0.3488182718553158) -- (0.453430058077434,0.35613165918762113);
\draw[color=qqqqff]  (0.453430058077434,0.35613165918762113) -- (0.4662832883182421,0.36170551987479616);
\draw[color=qqqqff]  (0.4662832883182421,0.36170551987479616) -- (0.47913651855905026,0.3655113276404127);
\draw[color=qqqqff]  (0.47913651855905026,0.3655113276404127) -- (0.4919897487998584,0.3675300434948423);
\draw[color=qqqqff]  (0.4919897487998584,0.3675300434948423) -- (0.5048429790406666,0.367751710097094);
\draw[color=qqqqff]  (0.5048429790406666,0.367751710097094) -- (0.5176962092814748,0.3661752399858601);
\draw[color=qqqqff]  (0.5176962092814748,0.3661752399858601) -- (0.530549439522283,0.36280839246928404);
\draw[color=qqqqff]  (0.530549439522283,0.36280839246928404) -- (0.5434026697630912,0.3576679385984161);
\draw[color=qqqqff]  (0.5434026697630912,0.3576679385984161) -- (0.5562559000038994,0.3507800183016572);
\draw[color=qqqqff]  (0.5562559000038994,0.3507800183016572) -- (0.5691091302447076,0.34218069828352776);
\draw[color=qqqqff]  (0.5691091302447076,0.34218069828352776) -- (0.5819623604855158,0.33191674349755157);
\draw[color=qqqqff]  (0.5819623604855158,0.33191674349755157) -- (0.594815590726324,0.3200466185807298);
\draw[color=qqqqff]  (0.594815590726324,0.3200466185807298) -- (0.6076688209671322,0.3066417380748772);
\draw[color=qqqqff]  (0.6076688209671322,0.3066417380748772) -- (0.6205220512079404,0.29178798472518563);
\draw[color=qqqqff]  (0.6205220512079404,0.29178798472518563) -- (0.6333752814487486,0.2755875123134638);
\draw[color=qqqqff]  (0.6333752814487486,0.2755875123134638) -- (0.6462285116895568,0.25816084129202327);
\draw[color=qqqqff]  (0.6462285116895568,0.25816084129202327) -- (0.659081741930365,0.2396492387996857);
\draw[color=qqqqff]  (0.659081741930365,0.2396492387996857) -- (0.6719349721711732,0.22021734480067387);
\draw[color=qqqqff]  (0.6719349721711732,0.22021734480067387) -- (0.6847882024119814,0.2000559563253288);
\draw[color=qqqqff]  (0.6847882024119814,0.2000559563253288) -- (0.6976414326527896,0.17938480260544387);
\draw[color=qqqqff]  (0.6976414326527896,0.17938480260544387) -- (0.7104946628935978,0.15845502253377275);
\draw[color=qqqqff]  (0.7104946628935978,0.15845502253377275) -- (0.723347893134406,0.13755087643381564);
\draw[color=qqqqff]  (0.723347893134406,0.13755087643381564) -- (0.7362011233752142,0.11698996934329349);
\draw[color=qqqqff]  (0.7362011233752142,0.11698996934329349) -- (0.7490543536160224,0.09712091998784124);
\draw[color=qqqqff]  (0.7490543536160224,0.09712091998784124) -- (0.7619075838568306,0.07831698361010261);
\draw[color=qqqqff]  (0.7619075838568306,0.07831698361010261) -- (0.7747608140976388,0.06096368027521411);
\draw[color=qqqqff]  (0.7747608140976388,0.06096368027521411) -- (0.787614044338447,0.045438147245458735);
\draw[color=qqqqff]  (0.787614044338447,0.045438147245458735) -- (0.8004672745792552,0.032078068949969306);
\draw[color=qqqqff]  (0.8004672745792552,0.032078068949969306) -- (0.8133205048200634,0.02113928899933107);
\draw[color=qqqqff]  (0.8133205048200634,0.02113928899933107) -- (0.8261737350608716,0.012744584002924183);
\draw[color=qqqqff]  (0.8261737350608716,0.012744584002924183) -- (0.8390269653016798,0.006832683193817566);
\draw[color=qqqqff]  (0.8390269653016798,0.006832683193817566) -- (0.851880195542488,0.0031264690509556926);
\draw[color=qqqqff]  (0.851880195542488,0.0031264690509556926) -- (0.8647334257832962,0.0011484345461180913);
\draw[color=qqqqff]  (0.8647334257832962,0.0011484345461180913) -- (0.8775866560241044,0);
\draw[color=qqqqff]  (0.8775866560241044,0) -- (0.8904398862649126,0);
\draw[color=qqqqff]  (0.8904398862649126,0) -- (0.9032931165057208,0);
\draw[color=qqqqff]  (0.9032931165057208,0) -- (0.916146346746529,0);
\draw[color=qqqqff]  (0.916146346746529,0) -- (0.9289995769873372,0);
\draw[color=qqqqff]  (0.9289995769873372,0) -- (0.9418528072281454,0);
\draw[color=qqqqff]  (0.9418528072281454,0) -- (0.9547060374689536,0);
\draw[color=qqqqff]  (0.9547060374689536,0) -- (0.9675592677097617,0);
\draw[color=qqqqff]  (0.9675592677097617,0) -- (0.98041249795057,0);
\draw[color=qqqqff]  (0.98041249795057,0) -- (0.9932657281913781,0);
\draw[color=qqqqff]  (0.9932657281913781,0) -- (1.0061189584321863,0);
\draw[color=qqqqff]  (1.0061189584321863,0) -- (1.0189721886729945,0);
\draw[color=qqqqff]  (1.0189721886729945,0) -- (1.0318254189138027,0);
\draw[color=qqqqff]  (1.0318254189138027,0) -- (1.044678649154611,0);
\draw[color=qqqqff]  (1.044678649154611,0) -- (1.0575318793954191,0);
\draw[color=qqqqff]  (1.0575318793954191,0) -- (1.0703851096362273,0);
\draw[color=qqqqff]  (1.0703851096362273,0) -- (1.0832383398770355,0);
\draw[color=qqqqff]  (1.0832383398770355,0) -- (1.0960915701178437,0);
\draw[color=qqqqff]  (1.0960915701178437,0) -- (1.108944800358652,0);
\draw[color=qqqqff]  (1.108944800358652,0) -- (1.1217980305994601,0);
\draw[color=qqqqff]  (1.1217980305994601,0) -- (1.1346512608402683,-0.0010870931101085687);
\draw[color=qqqqff]  (1.1346512608402683,-0.0010870931101085687) -- (1.1475044910810765,-0.0029968547226134705);
\draw[color=qqqqff]  (1.1475044910810765,-0.0029968547226134705) -- (1.1603577213218847,-0.006608089949889535);
\draw[color=qqqqff]  (1.1603577213218847,-0.006608089949889535) -- (1.173210951562693,-0.01240632853642768);
\draw[color=qqqqff]  (1.173210951562693,-0.01240632853642768) -- (1.1860641818035011,-0.02067913398213449);
\draw[color=qqqqff]  (1.1860641818035011,-0.02067913398213449) -- (1.1989174120443093,-0.031497642424582056);
\draw[color=qqqqff]  (1.1989174120443093,-0.031497642424582056) -- (1.2117706422851175,-0.044746787757087146);
\draw[color=qqqqff]  (1.2117706422851175,-0.044746787757087146) -- (1.2246238725259257,-0.06017588228235131);
\draw[color=qqqqff]  (1.2246238725259257,-0.06017588228235131) -- (1.237477102766734,-0.07745010580542122);
\draw[color=qqqqff]  (1.237477102766734,-0.07745010580542122) -- (1.2503303330075421,-0.09619341174436043);
\draw[color=qqqqff]  (1.2503303330075421,-0.09619341174436043) -- (1.2631835632483503,-0.11602013531266113);
\draw[color=qqqqff]  (1.2631835632483503,-0.11602013531266113) -- (1.2760367934891585,-0.13655609546440745);
\draw[color=qqqqff]  (1.2760367934891585,-0.13655609546440745) -- (1.2888900237299667,-0.1574513092846422);
\draw[color=qqqqff]  (1.2888900237299667,-0.1574513092846422) -- (1.301743253970775,-0.17838660823440453);
\draw[color=qqqqff]  (1.301743253970775,-0.17838660823440453) -- (1.3145964842115831,-0.19907612521865917);
\draw[color=qqqqff]  (1.3145964842115831,-0.19907612521865917) -- (1.3274497144523913,-0.21926716619110595);
\draw[color=qqqqff]  (1.3274497144523913,-0.21926716619110595) -- (1.3403029446931995,-0.23873855079359146);
\draw[color=qqqqff]  (1.3403029446931995,-0.23873855079359146) -- (1.3531561749340077,-0.25729815917713855);
\draw[color=qqqqff]  (1.3531561749340077,-0.25729815917713855) -- (1.366009405174816,-0.27478016336508465);
\draw[color=qqqqff]  (1.366009405174816,-0.27478016336508465) -- (1.3788626354156241,-0.2910422388583709);
\draw[color=qqqqff]  (1.3788626354156241,-0.2910422388583709) -- (1.3917158656564323,-0.3059629284235134);
\draw[color=qqqqff]  (1.3917158656564323,-0.3059629284235134) -- (1.4045690958972405,-0.31943924911622507);
\draw[color=qqqqff]  (1.4045690958972405,-0.31943924911622507) -- (1.4174223261380487,-0.33138458266411985);
\draw[color=qqqqff]  (1.4174223261380487,-0.33138458266411985) -- (1.430275556378857,-0.34172685871104996);
\draw[color=qqqqff]  (1.430275556378857,-0.34172685871104996) -- (1.4431287866196651,-0.35040702323191164);
\draw[color=qqqqff]  (1.4431287866196651,-0.35040702323191164) -- (1.4559820168604733,-0.3573777759103421);
\draw[color=qqqqff]  (1.4559820168604733,-0.3573777759103421) -- (1.4688352471012815,-0.3626025572363044);
\draw[color=qqqqff]  (1.4688352471012815,-0.3626025572363044) -- (1.4816884773420897,-0.36605476642275553);
\draw[color=qqqqff]  (1.4816884773420897,-0.36605476642275553) -- (1.494541707582898,-0.36771719360607885);
\draw[color=qqqqff]  (1.494541707582898,-0.36771719360607885) -- (1.5073949378237061,-0.3675816533317716);
\draw[color=qqqqff]  (1.5073949378237061,-0.3675816533317716) -- (1.5202481680645143,-0.3656488105114961);
\draw[color=qqqqff]  (1.5202481680645143,-0.3656488105114961) -- (1.5331013983053225,-0.3619281945533952);
\draw[color=qqqqff]  (1.5331013983053225,-0.3619281945533952) -- (1.5459546285461307,-0.3564384020172979);
\draw[color=qqqqff]  (1.5459546285461307,-0.3564384020172979) -- (1.558807858786939,-0.3492074927858411);
\draw[color=qqqqff]  (1.558807858786939,-0.3492074927858411) -- (1.5716610890277471,-0.3402735892231526);
\draw[color=qqqqff]  (1.5716610890277471,-0.3402735892231526) -- (1.5845143192685553,-0.32968569190396696);
\draw[color=qqqqff]  (1.5845143192685553,-0.32968569190396696) -- (1.5973675495093635,-0.31750472889632386);
\draw[color=qqqqff]  (1.5973675495093635,-0.31750472889632386) -- (1.6102207797501717,-0.3038048577062133);
\draw[color=qqqqff]  (1.6102207797501717,-0.3038048577062133) -- (1.62307400999098,-0.28867503892761287);
\draw[color=qqqqff]  (1.62307400999098,-0.28867503892761287) -- (1.6359272402317881,-0.27222089693709517);
\draw[color=qqqqff]  (1.6359272402317881,-0.27222089693709517) -- (1.6487804704725963,-0.2545668733851066);
\draw[color=qqqqff]  (1.6487804704725963,-0.2545668733851066) -- (1.6616337007134045,-0.23585866036838699);
\draw[color=qqqqff]  (1.6616337007134045,-0.23585866036838699) -- (1.6744869309542127,-0.21626586698813213);
\draw[color=qqqqff]  (1.6744869309542127,-0.21626586698813213) -- (1.687340161195021,-0.19598481824855635);
\draw (-2,0.03) -- (-2,-0.03) node[below]{\tiny{$-2$}};
\draw (-1,0.03) -- (-1,-0.03) node[below]{\tiny{$-1$}};
\draw (0,0.03) -- (0,-0.03) node[below right=-1]{\tiny{$0$}};
\draw (1,0.03) -- (1,-0.03) node[below]{\tiny{$1$}};
\draw (2,0.03) -- (2,-0.03) node[below]{\tiny{$2$}};
\node at (2.48,0) [below]{\scriptsize{$t$}};
\end{axis}
\node at (2.63,2.3) [left]{\scriptsize{$y$}};
\node at (3.9,1.9) [blue, left]{\scriptsize{$\chi(t)$}};
\end{tikzpicture}
\end{document}