Existence of neighborhood monotonic increasing

Question

I was thinking about the following statement:

Supose $f\in C^1$ a function s.t. $f(b)>f(a) \forall b\in (a,a+\epsilon)$. So, there's exist a neighborhood $(a,a+\delta)$ s.t. $f$ is monotonic increasing on that.

Is it true? Thank you in advance.

Answer 1

Not necessarily. Think of the function $f(x)=x^3\left(2+\sin\frac{1}{x}\right)$. You can define the function in 0 as $f(x)=0$. In the photo, the function is in green (it´s obviously $>0$ in $(0,\infty)$), but its derivative, in red, has negative values at any neighborhood of 0.

Also, the function is $C^1$. That is obvious at any point different than 0, and using the definition of derivative you can check that the derivative of the function at 0 is 0. Moreover, as can be seen in the picture (you can also check it yourself), the derivative of the function tends to 0 in 0, so the function has a continuous derivative.