# Implications of continuous differentiability at a point

Consider a function $f:\mathbb{R}^l\rightarrow \mathbb{R}$ continuously differentiable at $x_0$. This implies that

(1) the function is differentiable on a neighbourhood of $x_0$ which means that the function is continuous on a neighbourhood of $x_0$

(2) the partial derivatives are all well-defined on a neighbourhood of $x_0$.

Are (1) and (2) correct?