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Let $A \subset \mathbb{R^n}$ with $a$ being an interior point of $A$ let $f:A \to \mathbb{R}$. Suppose that ${\partial f \over \partial x_1}, ..., {\partial f \over \partial x_n}$ exist and are bounded in the neighbourhood of $a$ with some radius $r > 0$. Prove that $f$ is continuous at $A$.

I've seen many proof using the mean value theorem. But my question is, doesn't the mean value theorem require continuity?

Phantom
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1 Answers1

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You probably saw the proof in Bounded partial derivatives imply continuity

It indeed uses mean value theorem, which indeed requires continuity and even differentiability... but not of $f$, of its restriction to lines parallel to coordinate axes. Such a restriction is continuous, because it is differentiable, because the partial derivatives of $f$ are assumed to exist.

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