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

Question

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch 5.1

Questions about this corollary

(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?

2. '$\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

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

• Context is 2 exercises (Exer 4.25: Prove Cor 4.20 to Cauchy's Thm using Green's Thm assuming $f'$ is continuous.) and (Exer 4.38), an exercise on proving Cauchy's Integral Formula on a convex region.

• It seems like '$f'$ is continuous' a new assumption, but with Ch5 machinery, we can now show that we can deduce that $f'$ is continuous from the holomorphicity of $f$.

• I think the answer is yes if my understanding for 1 and 2 are right.

Answer 1

1. Yes. See def of holomorphicity and regions. This is even true for 'an open subset' instead of 'a region'. It's indeed probably used for emphasis because one part of the consequent is infinite differentiability and the other part is about continuity of partial derivatives (rather than partial holomorphics, if there's even such a thing).

2. Yes assuming the formula is right, but even without the formula, just refer to the text in red box in the paragraph before Cor 5.5: any sequence of partial derivatives is a constant times something differentiable, and thus a constant times something continuous.

3. Yes. Speaking of new assumptions, Exer 4.25 seems to be missing positively oriented which Green's Thm requires but which Cor 4.20 to Cauchy's Thm does not assume or not.

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