very nice, but a little snag: is not always true with these conditions.

yes, it is fine to show that , this means is continuous atSince , we can conclude that is continuous at .

Is that correct, and if not, how do I fix it?

you need to show the definition holds for all points in the domain of the function (which happens to be all real numbers). so let be an arbitrary point in the domain. you must show:Now, what if I need to prove that is continuous for all x. How would I find an appropriate delta then?

For every , there exists a such that and implies

to find your delta, you would use similar manipulations to what you have done so far