multivariable calculus - Missing continuity condition in theorem?

I'm going through the proof that all partials continuous $\implies$ $f$ is differentiable. Here's what my book says:

What I'm wondering about is how we can use the mean value theorem in step $2$. Doesn't the MVT require $\mathbf f$ (or its components $f_i$) be continuous on $[x_{k-1},x_k]$ for all $k$? That's not a part of the suppositions for this theorem. Did the author just forget to add that $\mathbf f$ needs to be continuous on $U$ or is there something I'm missing?

Answer

The mean value theorem is applied to the real function

$$t \mapsto f(x+\sum_{i=1}^{k-1}h_ie_i+te_k),$$
which is continuous since it is differentiable, as its derivative is given by the partial derivative of the function $f$ (just apply the definition of derivative).

For your edit, what he is using is the fact that $\Vert h_k \Vert \leq \Vert h \Vert$, and then putting $\Vert h \Vert$ in evidence.