Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $x_0 \in mathbb{R}^n$. Write $f=(f_1,...,f_m)$. Then if the function $\frac{\partial f_i}{\partial x_j} : \mathbb{R}^n \rightarrow mathbb{R}$ exists in a neighbourhood around $x_0$ and is continuous on that neighbourhood, then $f$ is differentiable at $x_0$?

Since the Jacobian matrix of $f$ at $x_0$, denoted by $f^{\prime}(x_0)$, is $$\big( \frac{\partial f_i}{\partial x_j}(x_0) \big) \in M_{m \times n}(\mathbb{R})$$ provided that all of partial derivatives are continuous.

I think this works because we can consider the functions $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ because $f$ outputs vectors in $\mathbb{R}^n$ we can write $f = (f_1,...,f_m)$ where $f_i : \mathbb{R}^n \rightarrow \mathbb{R}$ for $i = 1,...,m$. Since each of these functions $f_i$ are scalar-valued, we can compute their partial derivatives.