Lipschitz

A function $f$ is $L$-Lipschitz continuous if changing the input by a little bit can't change the output by more than a proportional amount,

for all $x, y$ in its domain. In particular, choosing $y = x + \epsilon$ implies that the gradient $\| \nabla f(x) \|$ is bounded by $L$, but the definition doesn't require that $f$ be differentiable.

Sometimes people talk about "Lipschitz smoothness" or an "$L$-smooth" function to mean a function whose gradient is Lipschitz continuous with constant $L$. Interestingly, a function is Lipschitz smooth if its convex dual is strongly convex (see, e.g., https://xingyuzhou.org/blog/notes/Lipschitz-gradient).