You are wrong. In order for you to make sense of what you are saying, you first must REDEFINE f(x) to be f(x) = (x^2 - 1)(x - 1) when x != 1 and define f(1) = 2. Of course, then f will be continuous at x = 1 also.

A function is continuous at x = a if it is differentiable at x = a.

You do understand the concept, but your precision in the definitions is lacking.