Denote a differentiable function as $f(t): [0,+\infty) \to \mathbb{R}$. Let $\dot{f}(t)$ be the first derivative of $f(t)$. Besides, let $L_{2}$ and $L_{\infty}$ represent the function spaces of all square integrable functions and bounded functions respectively.
My question is that: if $f(t) \in L_{\infty}$ and $\dot{f}(t) \in L_{2} \cap L_{\infty}$, then is it possible to conclude that $f(t) \in L_{2}$?
I am quite new to this topic; hence, it would be great with anyone could suggest me a good reference to read. Thanks a lot!