Plotting a discontinuous function using pgfplots

Question

Say I want to plot the function |sin(x)| and its derivative, which has a discontinuity at x = 0, using pgfplots. How can I handle the discontinuity? What I'd like to happen is for the red line below to end at (0,-1), then jump to (0,1) and continue normally from there. My main issue here is having two points with the x-coordinate 0, I think.

I do not want to have two separate plots (as in this question, which is actually about division), as this would mess with automatic selection of the line style and other features of pgfplots (e.g. a single entry in the legend).

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If I read the documentation correctly, I could achieve the desired effect by explicitly providing the coordinates (0,-1), (0,nan), (0,1) and setting unbounded coords=jump, but I want to plot the function without specifying all the coordinates. Specifying the coordinate of the discontinuity is fine, of course.

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A starting point:

\documentclass{article}

\usepackage{pgfplots}

\begin{document}

\begin{tikzpicture}
  \begin{axis} [
        trig format plots=rad,
        domain=-pi:pi,
        samples=201,
        no markers,
        xtick={-pi, 0, pi},
        xticklabels={\(-\pi\), \(0\), \(\pi\)},
        ytick={-1, 0, 1},
        grid=major,
        typeset ticklabels with strut,
      ]
    \addplot {abs(sin(x))};
    \addplot {sign(x) * cos(x)};
  \end{axis}
\end{tikzpicture}

\end{document}

Answer 1

Here you go. Add unbounded coords=jump and plot (x==0?nan:sign(x) * cos(x)).

\documentclass{article}

\usepackage{pgfplots}
% \pgfplotsset{compat=1.17} %<-consider adding
\begin{document}

\begin{tikzpicture}
  \begin{axis} [
        trig format plots=rad,
        domain=-pi:pi,
        samples=201,
        no markers,
        xtick={-pi, 0, pi},
        xticklabels={\(-\pi\), \(0\), \(\pi\)},
        ytick={-1, 0, 1},
        grid=major,
        typeset ticklabels with strut,
        unbounded coords=jump
      ]
    \addplot {abs(sin(x))};
    \addplot {(x==0?nan:sign(x) * cos(x))};
  \end{axis}
\end{tikzpicture}

\end{document}

ADDENDUM: For the sake of "if its doable let's do it even if it is crazy", here is a version that adds some jump marks. This is a compromise between subverting the plot handler (which is possible but even crazier) and just adding public global macros (which I thing one really has to avoid, if possible).

\documentclass{article}

\usepackage{pgfplots}
\pgfplotsset{compat=1.17}
\makeatletter
\newcommand\pgfpush[1]{\pgfutil@pushmacro#1}
\newcommand\pgfpop[1]{\pgfutil@popmacro#1}
\makeatother

\begin{document}

\begin{tikzpicture}
  \begin{axis} [
        trig format plots=rad,
        domain=-pi:pi,
        samples=201,
        no markers,
        xtick={-pi, 0, pi},
        xticklabels={\(-\pi\), \(0\), \(\pi\)},
        ytick={-1, 0, 1},
        grid=major,
        typeset ticklabels with strut,
        unbounded coords=jump,
        jump threshold/.initial=1
      ]
    \addplot {abs(sin(x))};
    \edef\isfirstpoint{1}
    \pgfpush\isfirstpoint
    \addplot+[scatter,
     scatter/@pre marker code/.append code={%
            \pgfkeys{/pgf/fpu=true,/pgf/fpu/output format=fixed}%           
            \pgfpop\isfirstpoint                        
            \pgfmathsetmacro{\myx}{\pgfkeysvalueof{/data point/x}}%
            \pgfmathsetmacro{\myy}{\pgfkeysvalueof{/data point/y}}%
            \ifnum\isfirstpoint=0
              \pgfpop\mylasty
              \pgfmathtruncatemacro{\itest}{(abs(\mylasty-\myy)<\pgfkeysvalueof{/pgfplots/jump threshold}?0:1)}
              \ifnum\itest=1\relax
                \pgfpop\mylastx
                \draw[fill=white] 
                 (axis direction cs:\mylastx-\myx,\mylasty-\myy) circle[radius=2pt]
                 (axis direction cs:0,0) circle[radius=2pt];
              \fi   
            \fi
            \edef\isfirstpoint{0}%
            \pgfpush\isfirstpoint
            \edef\mylasty{\myy}%
            \pgfpush\mylasty
            \edef\mylastx{\myx}%
            \pgfpush\mylastx
            }
    ] {(x==0?nan:sign(x) * cos(x))};
  \end{axis}
\end{tikzpicture}

\end{document}

You could argue that subverting the plot handler is a better option because then the user does not have to explicitly specify a jump threshold. I do not have any good counter-argument except that I am too lazy.