Holomorphic function is continuous?

[jaci55555]

Prove that any f: D -> C(complex) which is holomorphic in D subset of C is continuous in D



f is holomorphic in D if it is differentiable at every c element of D.
A function is differentiable at c if lim(h->0) (f(c+h) - f(c))/h exists.



I know from reals that a function is only differential if it is continuous... but not for complex numbers.

I have no idea how to prove a function continuous without drawing a graph or using the epsilon-delta method. Please help!

 

[micromass]

So you know that a real differentiable function is continuous. How did you prove that? Doesn't the exact same proof carry over to this case?

 

[jaci55555]

I didn't prove it, it was given. But I will definitely go try find a proof for it. Any other help will be greatly appreciated.

 

[jbunniii]

The following observation will be useful:

$$f(y) - f(x) = \frac{f(y) - f(x)}{y - x} (y - x)$$

valid for $y \neq x$.

 

[holomorphic]

What can you say about the partial derivatives of a complex function in the neighborhood of a point where that function is not continuous?

 

[jaci55555]

What can you say about the partial derivatives of a complex function in the neighborhood of a point where that function is not continuous?

Nothing yet... what should i be able to say?

I tried it like this:

THere is an open D subset of C with c element of D and f holomorphic in D. F is holomorphic at every c element of D

lim(z-> c) = [f(z) - f(c)] = lim(z->c)[(f(x) - f(z))/(z-c) * (z-c)] = lim(z->c) [(f(z) - f(c))/(z-c)] * lim(z->c)[z-c] = f'(c) * 0 = 0

Therefore f is continuous in D