Let $f:\Omega\to\mathbb R$, $\Omega\subseteq\mathbb R^n$, a continuous function and of class $C^{(1)}(\Omega\setminus \{ x_0\})$, $x_0\in\Omega$. Suppose that the limits $$l_i=lim_{x\to x_0}\frac{\partial f}{\partial x_i}(x), i=1,...,n$$ exist. Show that $$l_i = \frac{\partial f}{\partial x_i}(x_0)$$ and therefore $f\in C^{(1)}(\Omega)$. Hint: Use the mean-value theorem.
I don't even know how to start here... I'd be very thankful if someone could tell me how to approach this exercise.
Thanks!