A function $f: U\subseteq\mathbb{R^n}\to\mathbb{R^m}$ is continuously differentiable in some open set containing $\vec a$ if and only if all partial derivatives of $f$ are continuous in the same open set.
If yes, can someone help me go about the proof?
Surely not. How can continuity of partial derivative at a point give differentiability at other points? If all the partial derivatives are continuous throughout the open set then $f$ is continuously differentiable on that open set (and conversely).