Strictly increasing function implies positive derivative?

Question

We know that a if function $f:[a,b]→R$ is continuous on $[a,b]$ and $f′>0$ on $(a,b)$, except at a finite number of points in $(a,b)$, $f$ is strictly increasing on $[a,b]$. Is there any counterexample that shows the converse fails?

It's given that f is continuous. I want to know, if the function is increasing strictly, does this mean that the derivative of the function is positive everywhere except, possibly at a finite number of points in (a,b)? — theduckgoesquark, Dec 07 '17 at 22:29

score 3 · Answer 1 · answered Dec 07 '17 at 23:09

3

Hint: On $[0,1],$ consider the function

$$f(x) = \int_0^x t\sin^2 (1/t)\, dt.$$

answered Dec 07 '17 at 23:09

zhw.

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Strictly increasing function implies positive derivative?

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