How rigorous does this need to be? For instance, are you suppose to use the epsilon-delta definitions? It is very easy to apply in this case:
For all , choose so that . So the limit is .
You can easily generalise this to arbitrary instead of . And by generalise I mean replace every instance of with and you'll get a valid proof.
The limit for b) is also two. You need minor modification to the proof above. For example, take to be the distance for (or ) to the nearest integer.
In light of parts a) and b), it should be obvious if and where the function is continuous.