# Maximum number of times a function crosses a horizontal line

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.

• I would guess that it is a bit too general to have a specific name? – copper.hat Jun 30 '19 at 17:47

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.