Simple humps of a continuous function Suppose $y=f(x)$ is a continuous function and $f(x)=f(x')$ with $x≠x'$. Can we always find a sub-interval of the interval $[x, x']$ where $f$ is a simple hump or trough? By a simple hump, I mean a curve that rises monotonically from a certain height $y=k$, reaches a maximum, and then falls monotonically back to $y=k$. A simple trough is the inverse of that.

No, we can't necessarily do that. Take, for instance, the Weierstrass function, whose graph is a fractal, going up and down infinitely many times on any interval.

Note that if your function is indeed a bumb, meaning that it is first increasing then decreasing, then it is necessarily differentiable almost everywhere.