The function *f* will be discontinuous at any value of *c* where *f*(*c*) is undefined, where does not exist, or where . Think of such values as problem spots. We need to look at the function's definition and find these "problem spots". Where is *f* undefined? It looks like *x* = 2 could be trouble, since then However, when *x* = 2 we're not using this part of the function's definition. We only use when *x* < 0. Since 2 > 1, we have *f*(2) = 4. The function *f* is undefined at . When we're using the piece of the function definition that says so Therefore *f* is discontinuous at . Where can not exist? This can happen in between the "pieces". That is, where the function definition changes, or where individual pieces have undefined limits. The only place an individual piece of this function has an undefined limit is at , which we've already taken care of. For this function, the function definition changes at 0 and 1. We need to investigate the limit of *f* as *x* approaches each of these values. First off, does exist? For the left-sided limit, we find For the right-sided limit, we find Since the one-sided limits disagree, does not exist, and *f* is discontinuous at *x* = 0. Does exist? For the left-sided limit, we find For the right-sided limit, we find Since the one-sided limits agree, Finally, we need to see if *f*(*c*) and disagree. Again, we only need to inspect those values of *c* where the function's definition changes over. Since the function *f* is continuous at 1. We have no more possible trouble spots, *f* is discontinuous only at and *x* = 0. |