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?
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.