Consider the function
Determine what happens to as . Define so that will be continuous at , if possible.
First, we sketch the graph of on the intervals and .
To determine what happens to as we can take the limit, (recalling that ),
Since is not defined at it is not continuous there. However, since the limit exists we could redefine by
Then, this new (which has the same values as the original function everywhere the original function is defined, but has the additional feature of being defined at ) is continuous at since it is defined there and .
Note what we have done here. We had a function which was undefined at the point . We created a new function (which we also called , which is a bit confusing) that took the same values as the original function for all , and took the value 1 when . In this way we “removed” the discontinuity of the original at the point , but we should be aware that this is actually a new function (since it has a different domain than the original).