Are 1/x and 1/x^2 continuous functions?
Answer:
No, because when x=0 there is a whole on the graph, and for a function to be continuous there can not be a whole or an asymptote.
No, there are 3 steps to determine continuity

The function at a certain value must be defined
>1/x and 1/x^2 fail that test at x=0 
The limit must approach the same value on both sides
>1/x approaches inf. and inf at x=0
>1/x^2 technically approach infinity from both sides at x=0, but infinity isn’t a number, so 1/x^2 fails that test as well 
The limit must be equal to the function value
>You can already see how this is automatically untrue for both functions since they’re both undefined at x=0.
If a function fails ANY of these tests, the function isn’t continuous. (And 1/x and 1/x^2 fail all these steps)
TL;DR: No, because there’s an asymptote at x=0