I have the following question that I want to prove or disprove.
Suppose $F:[0,1]\rightarrow [0,1]$ be a non-constant absolutely continuous function. Then, there must exist an open interval in which $F$ is strictly monotone.
2026-02-23 15:30:45.1771860645
Does there exists an interval in which a non-constant absolutely continuous function is strictly monotone?
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This is false. There are examples of differentiable functions with bounded derivative who are nowhere monotone. These functions would be Lipschitz and so absolutely continuous. The reference is a paper of Y. Katznelson and K. Stromberg. "Everywhere differentiable, nowhere monotone, functions, Am. Math. Monthly, 81 (4), (1974), 349-353. See math overflow.