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We know that continuous nowhere monotonic functions exist (e.g., the Weierstrass function), and that there are differentiable functions with pathological properties. But this raised the following question in my mind:

Does there exist a real-valued function $f: \mathbb{R} \to \mathbb{R}$ that is:

Differentiable almost everywhere (i.e., differentiable at all points except a set of measure zero), and Nowhere monotonic on any nontrivial interval?

Such a function would be quite pathological: smooth "almost everywhere" yet resisting monotonic behavior completely on any interval.

What I tried:

  • Explored classical constructions like the Weierstrass function, which is continuous everywhere but differentiable nowhere—not suitable here.
  • Considered functions of bounded variation, but they tend to be monotonic on subintervals, which disqualifies them.
  • Reviewed functions with absolutely continuous derivatives, but they are monotonic on subsets if their derivative is of one sign—even locally.
Martin Sleziak
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F. A. Mala
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  • Do you want it continuous? It is easier of you don't require it to be continuous, I think. – Thomas Andrews Jun 12 '25 at 15:50
  • @mihaild Ah yes. If you have $f'(x)>0,$ you definitely have an open interval of $x$ where $f(y)<f(x)$ when $y<x,$ and $f(x)<f(y)$ when $x<y.$ But that doesn't mean for any $y_1<y_2$ in the interval, that $f(y_1)\leq f(y_2).$ Jumped to a conclusion. – Thomas Andrews Jun 12 '25 at 17:57

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