I'm interested in non-trivial examples of a continuously differentiable function $f: \mathbb{R}^d \to \mathbb{R}$ such that both $f$ and its gradient $\nabla f$ are both globally Lipschitz.
I could come up with trivial examples where $f$ is constant or affine. A more interesting example is where $f$ is affine only outside of a bounded set and finite inside (e.g., the 1d function $f(x) = x^2$ for $|x| \le 1$ and $2 |x| - 1$ otherwise), but I still think that this example is pathological.
In 1D: any function $f$ such that $f$, $f'$, $f''$ are bounded will satisfy the requirements thanks to the MVT. Such natural examples are $\arctan, \sin, \cos\ldots$
In nD, in the same spirit: $\exp(-\sum x_i^2)$ for instance.