Continuity of 2-branch function, point offset: where

Determine whether this function is continuous at x = 2. If it is discontinuous, is the discontinuity removable?

We can use the Definition of Continuity:

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

Does the function have a limit at the same point? Yes: see the related problem.

Is the function continuous? Since the limit is 5 at x = 2, but the function has a different value there, we conclude that the function is discontinuous at x = 2.

Is the discontinuity removable?
Yes, the function could be redefined to have the value 5 at x = 2.