 If we have a discontinuous function, we can still determine some information about them, but not quite as much, and not with as much certainty. For example, suppose I have a function that's discontinuous at x equals 7, but is continuous everywhere else in the interval between 0 and 10. What we'll see is this one little discontinuity at x equals 7 makes it very much more difficult to find anything about the function. So the first thing we might want to find out is where do we have solutions to f of x equals 0? And we might proceed as follows. Since f of x is continuous except at x equals 7, then the intermediate value theorem does apply for any interval that does not contain x equals 7. And so if I want to find f of x equals 0, I might notice that at x equals 2, f of 2 equals 5, which is too big, at x equals 4, f of 4 is negative 3, which is too small. And since f of x is continuous in this interval between 2 and 4, then I know that some place in between is going to be just right. So f of c will be 0 for some value in this interval. What about this other interval? We see that f of 6 is negative 5, which is too small, and f of 7 is 5, which is too big, but f of x is not continuous in the interval between 6 and 7 because it's discontinuous at x equals 7. And so the intermediate value theorem tells us absolutely nothing about whether or not a solution to f of x equals 0 exists in this interval. What about the limit as x approaches 4 of f of x? So we do know that f of x is continuous at x equals 4. And since we know the definition of continuity, we know that the limit as x approaches 4 of f of x is going to be f of 4. And we know f of 4 is negative 3. And so that tells us the limit as x approaches 4 of f of x will be negative 3. Let me say, great, the limit is the function value. Well, why is the limit as x approaches 7 of f of x not equal to 5? So let's take this problem apart. Since f of x is not continuous at x equals 7, then we know either f of 7 does not exist, but we know that f of 7 is equal to 5. So that's not a problem. The limit as x approaches 7 does not exist. Well, if the limit doesn't exist, then it's not equal to 5. And the other possibility for being not continuous is that the limit is not equal to the function value. So even if the limit does exist, whatever it is can't be equal to 5. And so it's useful to contrast this to the previous problem where we were continuous, we were able to find the limit immediately, where we are not continuous, we have no idea what the limit is, except we know what it's not equal to.