For the function, determine all values at which the function is discontinuous.

Answer

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.