Just needed a double check on my working to see if I am on the right track!

Prove that is continuous at .

Given , we have , provided that

Therefore, .

Is this correct?

And I have another question about continuity!

Let denotes the set of rational numbers, consider the function

(i) Prove that g(x) is continuous at x=0.

(ii) State the least upper bound for g on the interval [0,1] and state whether g attains a maximum on [0,1].Give clear explanations to justify your statements.