# Extended functions continuous at $z=(0,0)$

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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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.
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