How to prove that the gradient of nonconvex smooth function is Lipschitz continuous?

Question

Given a 1-smooth and nonconvex function $f(x)$, satisfy $$f(y) - f(x) -\left<\nabla f(x),y-x\right>\leq \frac{L}{2}\|x-y\|^2$$ How to prove that the gradient is Lipschitz continuous, i.e., $$\|\nabla f(x) - \nabla f(y)\|\leq L\|x-y\|$$

Answer 1

Every concave, smooth function $f$ satisfies your first inequality for $L = 0$. Hence, your claim is not true.

Answer 2

The statement is not true as @gerw has pointed out, just take any concave function as a counter-example. The converse, however, is true. See here for a proof.