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There are 4 functions:

$$\frac{Re(z)}{|z|},\frac{z}{|z|},\frac{Re(z^2)}{|z|^2},\frac{zRe(z)}{|z|}$$

I need to determine which of these functions can be defined at $z=0$ in such a way that the extended functions are continuous at $z=0$.

I know that if the extended functions are to be continuous at $z=0$, then $\displaystyle \lim_{z\to 0}f(z)$ must exist. However, I am having some difficulty determining the limits of these functions as $z$ approaches $0$. I can't figure out how to manipulate the current form of these functions so that they are not indeterminate.

I've tried substituting $z=x+iy$ into the expressions but with no avail.

I've also tried changing everything into the polar form, which leads to a lot of possible manipulations, but I'm not sure if I can define a $\theta$ when $z=(0,0)$.

I'd appreciate any guidance! I'm just not sure where to go from here.

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1 Answer 1

up vote 2 down vote accepted

Try with $z=x$ or with $z=iy$ with $x,y$ real. This will find the cases when you cannot extend the functions. In the last case you should take the modulus of the function and see that it goes to zero.

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That makes sense. To prove that $\displaystyle \lim _{z \to 0}f(z)=0$, should I use the definition of a limit? If so, I need to show that $\forall \epsilon, \exists \delta >0 \ni |z|<\delta \Rightarrow |\frac{zRe(z)}{|z|}|<\epsilon$. How would I manipulate that last portion so that I can find a suitable value for $\delta$? –  Jess Feb 8 '13 at 7:51
    
$\lim_{z\to 0} f(z) = 0$ is equivalent to $\lim_{z\to 0} \lvert f(z) \rvert = 0$. –  Emanuele Paolini Feb 8 '13 at 7:55
    
Use $\lvert zw \rvert = \lvert z\rvert \cdot \lvert w \rvert$ and $\lvert Re(z)\rvert \le \lvert z\rvert$. –  Emanuele Paolini Feb 8 '13 at 7:57
    
I forgot that $|Re(z)|\le|z|$. Thank you so much for your help! –  Jess Feb 8 '13 at 8:04

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