Let .

1.) Prove (using the definition) that at is continuous on .

2.) Prove (using the def. of differentiability) that at is differentiable.

Well, I don't know if this is helpful, but I know is not uniformly continuous on . There is a similar example in my book proving that x^2 is continuous on . They do:

Given :

Now they find an upperbound on by makings ure does not exceed 1. Apparently this insures all values of will fall in the interval .

I don't entirely get that ste.

Then, they get:

They let be given, then choose , then

.

I'm assuming it's similar for ?