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 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 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 we find 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 anywhere. Again, we only need to inspect those values of *c* where the function 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. |