Discourage linebreaks before/after math

Question

I would like to discourage linebreaks right before and after inline math displays, such that

now let $f$ be the function that connects

acts similar to

now let~$f$~be the function that connects 

It's not important to me that the behaviour is exactly the same - in fact, I only want to discourage linebreaks at these locations, not necessarily completely forbid them. I have a long document with lots of inline math (as a matter of fact, my Masters thesis) and I would like to see things like

                      ... now let f
be the function that connects ...

and

                        ... now let
f be the function that connects ...

less often, without having to edit every single occurence (which would be several hundred manual edits).

Answer 1

Inserting a penalty after the math isn't so hard, see how the break after $h$ is suppressed.

\documentclass{article}

\begin{document}

one two three $f$ four five six
one two three $f$ four five six
one two three $h$ four five six
one two three $f$ four five six
one two three $f$ four five six
one two $g$ three $f$ four five six
one two three $f$ four five six
one two three $f$ four five six
one two three $f$ four five six


\def\hmmm{\nolinebreak[3]}
\everymath{\aftergroup\hmmm}

one two three $f$ four five six
one two three $f$ four five six
one two three $h$ four five six
one two three $f$ four five six
one two three $f$ four five six
one two $g$ three $f$ four five six
one two three $f$ four five six
one two three $f$ four five six
one two three $f$ four five six



\end{document}

Suppressing breaks before the math is likely to be fragile when using the $ syntax most things you could think of would break something....

Answer 2

This method works, but is fragile. Maybe Lua has a better solution by traversing the nodes.

---

You can redefine the dollar, so:

%! TEX program = pdflatex
\documentclass{article}
\begingroup
\ExplSyntaxOn
\catcode`\$=\active
\protected\gdef${
  \unless\ifmmode
    % ======== we're near the start of a formula
    \ifdim\lastskip>\c_zero_dim
      \l_tmpa_dim=\lastskip\relax
      \unskip
      \nolinebreak[3]
      \hskip\l_tmpa_dim
    \fi
    \(
  \else
    % ======== we're near the end of a formula ========
    \)\nolinebreak[3]
  \fi
}
\ExplSyntaxOff
\endgroup
\begin{document}

one two three $f$ four five six
one two three $f$ four five six
one two three $h$ four five six
one two three $f$ four five six
one two three $f$ four five six
one two $g$ three $f$ four five six
one two three $f$ four five six
one two three $f$ four five six
one two three $f$ four five six

\catcode`\$=\active

one two three $f$ four five six
one two three $f$ four five six
one two three $h$ four five six
one two three $f$ four five six
one two three $f$ four five six
one two $g$ three $f$ four five six
one two three $f$ four five six
one two three $f$ four five six
one two three $f$ four five six

\end{document}

Of course this will break things that rely on $ having the catcode it should have e.g. TikZ calc library:

\begin{tikzpicture}
  \draw (0, 0)--($(0, 0)+(1, 1)$);  % "Missing $ inserted." --- what?
\end{tikzpicture}

There are some ways to patch it e.g.

\AddToHook{env/tikzpicture/begin}{\catcode`\$=3\relax}

But then the text inside nodes don't get this applied (this is fragile!):

\begin{tikzpicture}
  \node [text width=10cm, above] {one two three $f$ four five six one two three $f$ four five six one a $f$ four five six one two three $f$ four five six};
  \node [text width=10cm, below] {one two three $f$ four five six one two three $f$ four five six one a~$f$~four five six one two three $f$ four five six};
\end{tikzpicture}

In this specific case you can get away with using TikZ' node font:

\begin{tikzpicture}[node font={\catcode`\$=\active}]
  \node [text width=10cm, above] {one two three $f$ four five six one two three $f$ four five six one a $f$ four five six one two three $f$ four five six};
  \node [text width=10cm, below] {one two three $f$ four five six one two three $f$ four five six one a~$f$~four five six one two three $f$ four five six};
\end{tikzpicture}