A function that crosses each horizontal line only finitely many times

Question

Consider a real function $f(x)$ (not necessarily continuous) defined on a finite interval. Given a constant $C$, divide the interval to sub-intervals such that, in every sub-interval, either $f(x)<C$ or $f(x)>C$ (where the points $f(x)=C$ are ignored). Let $N(f,C)$ be the smallest number of sub-intervals in such a division.

Informally, $N(f,C)$ is approximately the number of times that the function $y=f(x)$ "crosses" the horiznotal line $y=C$, where "crosses" means that it goes from being below the line to being above the line or vice versa.

For example:

If $f(x)=\sin(x)$ defined on the interval $[-\pi,\pi]$, then $N(f,0)=2$, since $f(x)$ is negative on $(\pi,0)$ and positive on $(0,\pi)$.
If $f(x)=x\cdot \sin(1/x)$ on the same interval then then $N(f,0)=\infty$, since this function crosses the line y=0 infinitely many times.

What term describes the functions for which $N(f,C)$ is finite for every $C$ and on any interval?

Do you expect it to have a name, or are wondering if it does? — David P, Dec 23 '14 at 08:38
@DavidPeterson This seems to me a very natural property to have in a function. So I expect that either (a) it has a name, so that I can say "Let $f$ be an XYZ function"; (b) it is implied by another, more basic property, so that I can say "Let $f$ be an ABC function. This implies that it is XYZ". — Erel Segal-Halevi, Dec 23 '14 at 08:44
@MatthewLevy I think the "finite $N$" property is independent of continuity. — Erel Segal-Halevi, Dec 23 '14 at 09:07
I know its not implied, I'm just wondering what exactly he was asking... — Matthew Levy, Dec 23 '14 at 09:09
@ErelSegalHalevi: assume that $f$ is an analytic function on $I$. So it is $g=f-C$: assuming that $N(f,C)=\infty$, then the zeroes of $g$ accumulate somewhere in $I$, say in $x_0$. Then all the derivatives of $g$ in $x_0$ are zero, so $g$ is a constant function. — Jack D'Aurizio, Dec 23 '14 at 09:16
It would be interesting to find conditional relations with the assumption that ${x\mid f(x)=C}$ is finite for all $C$. The latter functions can be called just finite. — Andrea Mori, Dec 23 '14 at 09:16
@MatthewLevy I ask about a property of general functions, not necessarily continuous. — Erel Segal-Halevi, Dec 23 '14 at 09:25
If $f$ is differentiable and if the derivative only has finitely many zeros, $N(f,C)$ will be finite for every $C$ by Rolles theorem. — PhoemueX, Dec 23 '14 at 09:29

score 3 · Accepted Answer · edited Apr 27 '16 at 15:49

3

Sorry, can't comment: Look up the Banach Indicatrix of a function and functions of bounded variation.

The Banach indicatrix of a function $f\colon D \to \mathbb{R}$ is defined by $N(f,y) = \#\{x \in D \mid f(x) = y\}$ which is very similar to (but not exactly the same as) your definition.

edited Apr 27 '16 at 15:49

Daniel Fischer

211,575

answered Dec 23 '14 at 10:06

Jimmy P.

46

Interesting, thanks! It seems that finite crossing-number implies bounded variation (if the function itself is bounded), but not vice versa. – Erel Segal-Halevi Dec 24 '14 at 06:00

A function that crosses each horizontal line only finitely many times

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