Need some help with complex functions... :(

• Jan 23rd 2011, 03:18 AM
sedam7
Need some help with complex functions... :(
hello :D
I'm having big problem with simple issue, so if any one is willing to guide me to the right approach I will be most grateful :D

my problem is with this task :D

"If complex function $f(z)$ is regular in the area $D\subset \mathbb{C}$ than it's also continuous on same defined area."

and I need to prove it :D but i can't even start it because if I assume that some function is regular that implies that function is continuous.... !?!??! I have no clue how to do this :D
• Jan 23rd 2011, 03:26 AM
FernandoRevilla
For every $z_0\in D$ :

$f(z)-f(z_0)=\dfrac{f(z)-f(z_0)}{z-z_0}\cdot (z-z_0)\;,\;(z\neq z_0)$

Taking limits when $z\rightarrow z_0$ :

$\displaystyle\lim_{z \to z_0}{(f(z)-f(z_0))}=f'(z_0)\cdot 0=0$

Now, you can conclude.

Fernando Revilla
• Jan 23rd 2011, 03:38 AM
sedam7
Thank you very very much :D
(I'm an idiot hehehehehe)
• Jan 23rd 2011, 03:44 AM
FernandoRevilla
Quote:

Originally Posted by sedam7
(I'm an idiot hehehehehe)

Welcome to my Club. :)

Fernando Revilla