# 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$exist and are continuous}}$' box $\to$ 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?
• 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.

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