I came across this question when I tried to solve a math problem, and my solution was heavily based on this statement above. I find it very intuitive because you can't draw a continous function from one point of the plane to another (with different y's) without having to draw the line such that it is possible to find a very small interval on which the function is strictly increasing/decreasing. My question is the following: is this statement provable (and how the proof would unfold) or even true? I asked my math teacher whether there is a theorem about this topic, and he couldn't remember one.
2026-04-12 16:00:24.1776009624
Is it possible to prove that for any non-constant continous real function there is an interval on which the function is strictly monotonous?
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