F is continuous over the closed interval from a to b, so this includes not just the points between a and b, but the endpoints as well, if and only if f is continuous over the open interval and the one-sided limits, let me write this, and the limit as x approaches a from the right of f of x is equal to f of a, and the limit as x approaches b from the left of f of x is equal to f of b. Now what's going on here? Well, it's just saying that the one-sided limit when you're operating within the interval has to approach the same value as the function. So for example, if we said the closed interval from negative seven to negative five, well, this one is still reasonable, just based on the picking up your pencil thing, you don't have to pick up your pencil, and what you would do is at the endpoint, and at negative seven, this function is just plain old continuous, but if it wasn't defined over here, it could still be continuous because you would do the right-handed limit towards it, and you'd say, okay, the right-handed limit is equal to the value of the function, and then at this endpoint, at the second endpoint, you'd say, okay, the left-handed limit is equal to the function, even if it wasn't defined here, even if the two-sided limit were not defined. And so we could actually look at an example of that. If we were looking at the interval from the closed, and you could have one side open, one side closed, but let's just do the closed interval from negative three to negative two. So notice I did not have to pick up my pencil.