Complex analysis continuity of functions

[saraaaahhhhhh]

Homework Statement

The functions Re(z)/|z|, z/|z|, Re(z^2)/|z|^2, and zRe(z)/|z| are all defined for z!=0 (z is not equal to 0)
Which of them can be defined at the point z=0 in such a way that the extended functions are continuous at z=0?

It gives the answer to be:
Only f(z)=zRe(z)/|z|, f(0)=0

I see that f(0)=0 in this case, but I don't see how this is proven or shown. I don't see why the last equation works and the first three don't.

The Attempt at a Solution

I'm completely unsure of how to do this, I would have something if I even knew where to start. Maybe I'm confused about what extended functions are?

A link to the book:
http://books.google.com/books?id=Oy...ver&dq=introductory+complex+analysis#PPA37,M1

 

[mighty2000]

it is important to have a clear understanding of the terminology and concepts being used. In this case, the term "extended functions" refers to the extension of a function to include values at points where it is not originally defined. In this problem, the original functions are only defined for z!=0, but the question is asking which of them can be extended to include a value at z=0 in a way that makes the extended function continuous at that point.

To show that f(z)=zRe(z)/|z|, f(0)=0 is the only function that can be defined in this way, we can use the definition of continuity. A function is continuous at a point if the limit of the function at that point exists and is equal to the value of the function at that point.

For the first three functions, Re(z)/|z|, z/|z|, and Re(z^2)/|z|^2, we can see that they are not continuous at z=0 because the limit of each of these functions as z approaches 0 does not exist. This can be seen by approaching 0 from different directions and getting different values for the limit.

For the last function, zRe(z)/|z|, we can use the definition of the derivative to show that it is continuous at z=0. The derivative of this function is given by f'(z) = Re(z) - |z|/|z| = Re(z)-1. As z approaches 0, Re(z) also approaches 0, so f'(z) approaches -1. This means that the limit of the function as z approaches 0 is equal to the value of the function at z=0, which is 0. Therefore, this function is continuous at z=0.

In conclusion, the only function that can be extended to include a value at z=0 in a way that makes the extended function continuous at that point is f(z)=zRe(z)/|z|, f(0)=0. This is because the limit of this function at z=0 exists and is equal to the value of the function at that point.