# Thread: Continuity of a complex-valued function

1. ## 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.

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

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

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

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

4. Originally Posted by deepakpc007
when $\displaystyle r\to 0~?$, $\displaystyle z=re^{\mathif{i}\theta}$ then $\displaystyle \frac{z\cdot\text{Re}(z)}{|z|}=\frac{(re^{\mathif{ i}\theta})(r\cos(\theta)}{r}$=infinity???I think so.
Oh my no.
Is this true: $\displaystyle r\not= 0$ so $\displaystyle \frac{(re^{\mathif{i}\theta})(r\cos(\theta)}{r}=(r e^{\mathif{i}\theta})(\cos(\theta))~?$

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

6. Originally Posted by deepakpc007
Yes thats true.How can i prove with this?
What is there to prove.
As $\displaystyle r\to 0$ the other factors are bounded, so what is the limit?

7. Originally Posted by Plato
What is there to prove.
As $\displaystyle r\to 0$ the other factors are bounded, so what is the limit?
Edit : As $\displaystyle r\to 0$ the other factors are bounded,means a finite limit.Is it so?