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This thread is meant to record a question that I feel interesting during my self-study. I'm very happy to receive your suggestion and comments.

Let $A \subset \mathbb{R}^{d}$ be open convex, $f: A \rightarrow \mathbb{R}$ convex.

  • Theorem 1: If all partial derivatives $$ \frac{\partial f}{\partial x_{i}}(a) \quad(i=1, \ldots, d) $$ exist at $a \in A$, then $f$ is Fréchet differentiable at $a$.

  • Theorem 2: If all partial derivatives exist at each point of $A$, then $f \in \mathcal C^1 (A)$.

Akira
  • 18,439

1 Answers1

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Proof of Theorem 1: Clearly, $f$ is locally Lipschitz on $A$. We need the following results.

Lemma 1: Let $X$ be a normed space, $A \subset X$ convex open, and $f: A \to \mathbb{R}$ convex continuous at $a\in A$. If the directional derivative $f^{\prime}(a) [v]$ exists for every direction $v \in X$, then $f$ is Gâteaux differentiable at $a$.

Lemma 2: Let $A \subset \mathbb{R}^{d}$ be open, $f: A \rightarrow \mathbb{R}$, and $a \in A$. Assume that $f$ is Lipschitz on some neighborhood of $a$. Then $f$ is Fréchet differentiable at $a$ if and only if $f$ is Gâteaux differentiable at $a$.

By Lemma 1, $f$ is is Gâteaux differentiable at $a$. By Lemma 2, $f$ is Fréchet differentiable at $a$.

Proof of Theorem 2: By Theorem 1, $f$ is Fréchet differentiable on $A$.

Lemma 3: Let $X$ be a normed space, $A \subset X$ convex open, and $f: A\to \mathbb R$ convex continuous. If $f$ is Fréchet differentiable at each point of $A$, then $f \in \mathcal{C}^{1}(A)$.

By Lemma 3, $\partial f$ is continuous. This completes the proof.

Akira
  • 18,439