Discussing the uniform continuity of the following:

1.

Claim: f is uniformly continuous on (-1,1)

Proof.

Given , pick , then for each with , we would have

Q.E.D.

2.

Claim: f is uniformly continuous.

Proof.

Given , pick , then for each with , we would have

Q.E.D.

3.

Claim: f is not uniformly continuous.

So far, I know that , which is not below above as x approaches to 0.

But is it possible to prove this by defintion?

Proof so far.

Pick , then with , I need to show that But I'm stuck in here, any hints?

Thanks!!!