the derivative of function with finite limit at zero

Question

Let $f\in C(\mathbb{R})$, and there is number $b< \infty$, such as $f \rightarrow b$, $x\rightarrow 0$. Is it possible to say anything about $\sup_{0\le t\le x} f'(t)$ as $x \rightarrow0$?

Answer 1

Try $f(x) = x \sin \left( \dfrac 1{x^2} \right)$ with $f(0) = 0$.

$f$ is continuous but for $x \not= 0$ you get $f'(x) = \dfrac 1x \cos \left( \dfrac 1{x^2} \right) + \sin \left( \dfrac 1{x^2} \right).$