This is Exercise 10.3.5 from Analysis Vol.1 by Terence Tao.

Give an example of a subset $X \subset \mathbb{R}$ and a function $f: X \to \mathbb{R}$ which is differentiable on $X$, is such that $f'(x)>0$ for all $x \in X$, but $f$ is not strictly monotone increasing.



Since Terence Tao is a great mathematician, we must use some care in handling this exercise. Everybody knows that a function with positive derivative on an interval is increasing. However, Tao is asking for both a set and a function.

The easiest example is $X=(0,1) \cup (2,3)$, and $$ f(x)=x \quad\text{for $x \in (0,1)$}, f(x)=x-100 \quad\text{for $x \in (2,3)$}. $$ This function has $f'>0$ on $X$ but nevertheless it is not strictly increasing on $X$.

  • $\begingroup$ Shame didn't think of this. I was expecting something more complicated, like a function on $\mathbb{Q}$ or something. Anyway, this is a correct answer, thanks. $\endgroup$ – Not an ID Jun 11 '14 at 11:54
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    $\begingroup$ I beg to differ about "easiest"; what about $f(x) = -1/x$ on the set $X$ of non-zero reals...? ;) $\endgroup$ – Andrew D. Hwang Jun 11 '14 at 12:07
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    $\begingroup$ Well, I was essentially thinking of the shape of $X$, but the concept of simplicity is not totally ordered :-) $\endgroup$ – Siminore Jun 11 '14 at 12:22

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