# What is the name of this property of a function

I'm trying to find the right vocab word to describe a concept:

In computational geometry, there's a concept of a polygon "monotone" with respect to a line. Which means that the polygon intersects the line and all lines parallel to it in at most two places. I want to find the equivalent concept for real valued functions. That is, I want a property that describes a function that intersects a given line and lines parallel to it (in this case I'm interested in lines of the form $f(x) = c$, for all values of $c$, ie: horizontal lines) in at most two places.

The obvious analog, monotone functions, isn't what I want at all, since it limits the first derivative to always have the same sign, and even allows it to go flat for a bit.

The most sophisticated example I can think of: $\sin(x)$ over the interval $[0, 2\pi)$. If the interval was increased at all in either direction, there'd exist a horizontal line that intersects the function in 3 places. But if it's limited to just $[0, 2\pi)$, $\sin(x)$ only ever crosses any given horizontal line twice or less. From this example, there doesn't seem to be any rules you can impose on the first or second (or higher) derivatives, which was my initial thought.

Motivation: I'm thinking it's possible to decompose an oscillating (but not necessarily periodic) function in to chunks like this to help numerical root finders. eg: $f(x)=\sin(x)+c$ (for all $c$) evaluated over all possible intervals exactly $2 \pi$ wide will have this property. So if you break $\sin(x)+c$ in to $2\pi$ sized chunks, you know each chunk has at most two roots. Smaller intervals (no matter how small) can still have up to two roots, so $2\pi$ is the largest interval that still guarantees this property.

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## 2 Answers

Well, if what you're specifically interested in is horizontal lines, then if the function were to intersect each horizontal line exactly twice, it would be a 'two-to-one' function (i.e. for each $y$ value there are two distinct values of $x$ with $f(x) = y$). I'm not sure whether there's a good name for 'at-most-two-to-one' functions.

As for conditions on derivatives: supposing your function is differentiable, surely you just want the first derivative (i.e. the slope) to change sign at most once in any 'piece'?

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Twice is the upper bound, but imagine if there's any local extrema present. Then there'd be just one intersection, but it would have a higher multiplicity. And the line could miss the function entirely (eg: sin(x) with the line y = 2). – Jay Lemmon Feb 23 '13 at 1:59
Yes, that's why I said what it actually is is 'at-most-two-to-one'. – Tara B Feb 23 '13 at 12:05
Though the 'missing the function entirely' part isn't relevant to the name, since one-to-one functions don't have to be onto (e.g. $e^x$). It's possible that 'two-to-one' actually means 'at-most-two-to-one', since a continuous function could never be 'truly' two-to-one. – Tara B Feb 23 '13 at 12:19

If a real-valued function $f$ is "quasi-convex", then the set $\{x: f(x) \le c\}$ is convex for each given $c$, and is therefore an interval. Similarly, $f$ is said to be "quasi-concave" if the set $\{x: f(x) \ge c\}$ is convex for each given $c$. Taken together, I think these two classes of functions are pretty close to what you want. In computational geometry, you could also define a polygon to be monotone if its intersection with every line is an interval.

Note that quasi-convex functions are not necessarily continuous. So, if you want a closer correspondence with polygons, you may want to focus on functions that are both quasi-convex (or quasi-concave) and continuous.

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