First, we will look for values of x at which g is undefined. x = 1 is fine, since g(1) = 2 by the second piece of the definition. x = 2 is not fine: the function definition says what should happen for x < 2 and 2 < x, but doesn't make any provision for what happens at x = 2. Therefore g is undefined, and therefore discontinuous, at x = 2.
Now we will look for places where the limit is undefined or disagreeable. Since is undefined, is also undefined and g is discontinuous when x = 1.
To summarize, the function g is discontinuous at x = 1 and x = 2.