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Given a function $f:\mathbb{R}\rightarrow \mathbb{R}$, define $g(x)= | \{y |f(y)=x\}|$. Syppose that $g(x)$ is finite for all $x$. Define $a= \sup_x g(x)$. In words, $a$ is the maximum number of times $f$ crosses a line parallel to the $x$ axis.

In my application, it is convenient to define $a$ as a measure of complexity of $f$. I wanted to see if this is an object that appears in other contexts as well. If it has a name, or appears in any literature, I'd very much appreciate pointers.

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  • $\begingroup$ I would guess that it is a bit too general to have a specific name? $\endgroup$ – copper.hat Jun 30 '19 at 17:47
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Assuming that your use of the absolute value notation represents cardinality (and not Lebesgue measure), your function $g(x)$ is the Banach indicatrix of the function $f.$ So what you're interested in is the (global) maximum value of the Banach indicatrix. Off-hand, I don't know of any nontrivial results connected with this specific aspect of the Banach indicatrix, but the Banach indicatrix is certainly a well known tool in real analysis and harmonic analysis.

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