# Continuity of a complex-valued function

• Apr 28th 2011, 09:09 AM
deepakpc007
Continuity of a complex-valued function
Define the function f(z)=z*Real Part(z)/|z|,z!= 0
0 ,z=0
Prove that f(z) is continuous in the entire complex plane.

Pls somebody help me with dis one......thanx in advance.
• Apr 28th 2011, 09:26 AM
Plato
Quote:

Originally Posted by deepakpc007
Define the function f(z)=z*Real Part(z)/|z|,z!= 0
0 ,z=0
Prove that f(z) is continuous in the entire complex plane.

If $z=re^{\mathif{i}\theta}$ then $\frac{z\cdot\text{Re}(z)}{|z|}=\frac{(re^{\mathif{ i}\theta})(r\cos(\theta)}{r}$.

Now what happens when $r\to 0~?$
• Apr 28th 2011, 09:51 AM
deepakpc007
Quote:

Originally Posted by Plato
If $z=re^{\mathif{i}\theta}$ then $\frac{z\cdot\text{Re}(z)}{|z|}=\frac{(re^{\mathif{ i}\theta})(r\cos(\theta)}{r}$.

Now what happens when $r\to 0~?$

when $r\to 0~?$, $z=re^{\mathif{i}\theta}$ then $\frac{z\cdot\text{Re}(z)}{|z|}=\frac{(re^{\mathif{ i}\theta})(r\cos(\theta)}{r}$=infinity???I think so.
• Apr 28th 2011, 09:58 AM
Plato
Quote:

Originally Posted by deepakpc007
when $r\to 0~?$, $z=re^{\mathif{i}\theta}$ then $\frac{z\cdot\text{Re}(z)}{|z|}=\frac{(re^{\mathif{ i}\theta})(r\cos(\theta)}{r}$=infinity???I think so.

Oh my no.
Is this true: $r\not= 0$ so $\frac{(re^{\mathif{i}\theta})(r\cos(\theta)}{r}=(r e^{\mathif{i}\theta})(\cos(\theta))~?$
• Apr 28th 2011, 10:02 AM
deepakpc007
Quote:

Originally Posted by Plato
Oh my no.
Is this true: $r\not= 0$ so $\frac{(re^{\mathif{i}\theta})(r\cos(\theta)}{r}=(r e^{\mathif{i}\theta})(\cos(\theta))~?$

Yes thats true.How can i prove with this?
• Apr 28th 2011, 10:12 AM
Plato
Quote:

Originally Posted by deepakpc007
Yes thats true.How can i prove with this?

What is there to prove.
As $r\to 0$ the other factors are bounded, so what is the limit?
• Apr 28th 2011, 10:23 AM
deepakpc007
Quote:

Originally Posted by Plato
What is there to prove.
As $r\to 0$ the other factors are bounded, so what is the limit?

Edit : As $r\to 0$ the other factors are bounded,means a finite limit.Is it so?