I know if a function $f: \mathbb{R}^n \to \mathbb{R}$ is $L$-smooth, i.e. its gradient $\nabla f$ is $L$-Lipschitz continuous, then it satisfies the following inequality for any $x, x_0 \in \mathbb{R}^n$: $$ \left| f(x) - \left( f(x_0) + \nabla f(x_0)^\top (x - x_0)\right) \right| \leq \frac{L}{2} \lVert x - x_0 \rVert_2^2 \text{.} $$

Here's my question: is the converse true? I know that, provided $f$ is convex, then we can conclude that $f$ satisfying the above inequality must be $L$-smooth. But what if $f$ is not necessarily convex? If the converse is false, can you give some counterexamples?