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I am interested in this question:

Find a differentiable convex function such that its derivative is not continuous.

I found out that we cannot find such function if its domain is $\mathbb{R}$, since every differentiable convex function $f\colon \mathbb{R} \to \mathbb{R}$ is continuously differentiable (as proved here).

Therefore we have to look for multivariable functions, but it is not an easy work.

Thank you very much.

Tien Kha Pham
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1 Answers1

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Suppose $\Omega\subset\mathbb{R}^n$ is an open convex set, and $f : \Omega\to\mathbb{R}$ is a differentiable convex function. Then $\nabla f$ is continuous on $\Omega$.

This is a theorem, for example, in Convex Analysis by Rockafellar, page 246.