Is the derivative of a Lipschitz continuous gradient function is a continuous vector function?

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 $x,y \in \mathbb{R}^n$ and $\forall x,y \in \textbf{dom} f$. Is $f'(x)$ is a continuous vector function $\forall x,y \in \textbf{dom} f$?

Answer 1

Yes, we can draw that conclusion. But the conclusion is a bit more stronger. The gradient vector is uniformly continuous. Just use the definition of continuity. For all $\epsilon>0$ there is a $\delta=\epsilon/\alpha>0$ such that $$ \|x-y\|<\delta=\epsilon/\alpha \implies \|f^\prime(x)-f^\prime(y)\|\leq \alpha\|x-y\|\leq \alpha \delta =\alpha\cdot \epsilon/\alpha=\epsilon $$