We have known that a convex function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is Frechet differentiable iff $f\in C^1$ (class of Frechet differentiable functions with continuous gradients). This motivates to the following question:
Could we construct a convex function $f\in C^1$ but $f\notin C^{1,1}$ (class of Frechet differentiable functions with locally Lipschitz gradients)?
Thank you for all guidance and references.
Best Answer
The map $f:\Bbb R\to \Bbb R$, $f(x)=\frac23\lvert x\rvert^{3/2}=\int_0^x \lvert t\rvert^{1/2}\operatorname{sgn}t\,dt$ is convex and $C^1$, but $f'(x)=\lvert x\rvert^{1/2}\operatorname{sgn}x$ is not Lipschitz continuous in any neighbourhood of $0$. More generally, integrate your favourite monotone increasing continuous function which is not locally Lipschitz and you'll obtain a counterexample in $\Bbb R$.