Continuity of "offset" function:

Determine whether this function is continuous at x = 1. If it is discontinuous, is the discontinuity removable?

We can use the Definition of Continuity:

Is the function defined at x = 1? Yes, it is defined as equal to 2 there.

Does the function have a limit at the same point? No: see the related problem.

Is the function continuous? Since the limit doesn't exist, the function is not continuous at x = 1.

Is the discontinuity removable?
No, because the limit doesn't exist.