If the set of values , for which a function has positive derivative , is dense then is the function increasing? Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $A:=${ $x \in \mathbb R :f'(x)>0$ } is dense  in $\mathbb R$ , then is it true that $f$ is an increasing function ? What  happens if $f'$ is also continuous  ?

The answer is no.
Take a look at this paper. The corollary on p. 353 ensures that for any two disjoint countable dense subsets $A, B \subset \mathbb{R}$, there exists an everywhere differentiable function $H$ such that $H'>0$ on $A$ and $H'<0$ on $B$. Note that this function is monotone on no subinterval of $\mathbb{R}$.
I should give credit to user JLT from the French maths forum www.les-mathematiques.net, who pointed out this article to me on this thread.
Edit: Other references for such a counter-example include the paper mentioned by Dave Renfro in the comments below, also A Second Course on Real Functions by Rooij and Schikhof, example 13.2 p.80 (thank you Seirios for mentioning this to me in the French thread).
