
Continuity Question
Okay so the question is:
Let by
for
Prove that for each , is a continuous function of
([MTEXR[/MTEX is the real numbers, I'm not sure how to get it to look right).
I am letting and then trying to find a so whenever I am stuck trying to find the delta what will work. I start with and try and simplify but I am not sure how to get to the point where I can determine delta. Any help appreciated.

Re: Continuity Question
I think what you want is: \mathbb{R}
Anyway, let's calculate . Note that is undefined, so can only be a continuous function of .
Note: The final simplification is only possible if . If , then is a constant function and obviously continuous. Similarly, if , you also have a constant function, which is obviously continuous. So, assume and .
Since , the function is odd. So, you only need to consider continuity for and by symmetry, you will find continuity for all .
Now, given , , and , you want to find that works. First of all, we are going to want to limit the possible range of . We know that the function is not continuous at , so .
Does this help at all?