I'm inclined to the view that there are no real values of b which make the function f continuous at x=2.

If x is close to 2, but not equal to 2, f(x) is close to -3/b. But f(2) = b.

If we try b positive, then -3/b is negative so these things are not close

If we try b negative, similar but opposite problem.

If we try b zero then f is not defined.