This is what i've done so far:Prove that is continuous.

The definition of continuity is

In this case, will be

Since ,

Thus:

Combining this chain of inequalities with the one before gives:

Therefore let

Hence continuity is proved. I think this is correct, I would just like someone to look over it and tell me if this is how they would have written it or if this is even remotely correct.

Here's my other attempt with sequences:

Let and let .

As , approaches from above and approaches from below .

Therefore:

The right and left hand limits are the same so the function is continuous .

My other query with this is that I have only shown that the function is continuous for one set of sequences. Is there a way of showing that the function is continuous for all sequences that converge to ?

Thanks in advance to anyone who posts!