Is a Lipschitz continuous gradient function is a Lipschitz function as well?

Question

Let $f(x)$ be a Lipschitz continuous gradient function, that is $$ \|f'(x)-f'(y)\| \leq \alpha \|x-y\| $$ where $\|\cdot\| $ is Euclidean norm and $x,y \in \mathbb{R}^n$. Can we prove that the function is a Lipschitz function, that is $$ |f(x)-f(y)| \leq L \|x-y\| $$ If so, is there any relationship between $\alpha$ and $L$?