Differentiable at a point with positive derivative implies increasing in neighborhood of point?
Answer to the above question is false and it's disproven with the example given by @Hagen von Eitzen. However later @Yibo Yang commented that a weaker statement holds ( https://math.stackexchange.com/a/3582247/1177828 ).
My doubt is that the statement by @Yibo Yang can also be disproven by @Hagen von Eitzen example, right ? For every $\delta >0$ we can find $n$ such that $\frac{1}{2n\pi}<\delta$. Then $f'(\frac{1}{2n\pi})< 0$. So function is not increasing in $(0,\delta)$ for any $\delta >0$.
I felt like I did something wrong, but where ?