How to prove a gradient is Lipschitz continuous?

I have read in textbooks the following phrase, "assuming the gradient is locally Lipschitz continuous...", but how does one prove that the gradient is locally Lipschitz continuous?

If we have a continuously differentiable function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, from looking at previous stackexchange questions on this topic it seems we can show the gradient is Lipschitz continuous via Taylor's formula, but was wondering if anyone has insight or can point to papers/resources that clarify this further. Thank you.