If f is differentiable in a region $G$, then f is infinitely differentiable in G, and all partials of $f$ are continuous

(Cor 5.5) If f is differentiable in a region $$G$$, then f is infinitely differentiable in G, and all partials of $$f$$ with respect to $$x$$ and $$y$$ exist and are continuous.

1. '$$\color{blue}{\text{f is differentiable in a region G}}$$'

Is this the same as '$$f$$ is holomorphic in a region $$G$$', but 'differentiable' is instead used for emphasis?

1. '$$\color{green}{\text{all partials of f with respect to x and y exist and are continuous}}$$'

Why are they continuous?

Here is what I tried. Pf: Firstly, $$f_x=f'$$, by Cauchy-Riemann, is continuous because $$f'$$ is differentiable because $$f$$ is infinitely differentiable. Then extend by induction that (from here) $$f_{x^{(k)}y^{(n-k)}} = \frac{1}{(-i)^{n-k}1^k}f^{(n)}$$ is continuous because $$f^{(n)}$$ is continuous because $$f^{(n)}$$ is differentiable. QED

1. Does holomorphicity of $$f$$ imply continuity of $$f'$$ by Cor 5.5?

Pf: Differentiable on region iff holomorphic on region (Again, 'region' can be replaced with 'open subset'. See (1)). Then $f'$ is differentiable and thus continuous. QED