Continuity over an interval

what we're going to do in this video is explore continuity over an interval but to do that let's refresh our memory about continuity at a point so we say that f is continuous when X is equal to C if and only if so I'm going to make these two way arrows right over here the limit of f of X as X approaches C is equal to f of C and when we first introduced this we said hey this looks a little bit technical but it's actually pretty intuitive think about what's happening the limit as X approaches C of f of X so let's say that f of X as X approaches C is approaching some value so if we approach if we approach from the left we're getting to this value we're approaching the right we're getting this value well in order for the function to be continuous I have to draw this function without picking up my pen well the value of the function at that point should be the same as the limit that's this is really just a more rigorous way of describing this notion of not having to pick up your pencil this notion of connectedness that you don't have any jumps or any discontinuities of any kind so that out of the way let's discuss continuity over intervals let me delete this really fast so I have space to work with so we say someone first talked about an open interval and then we're gonna talk about a closed interval because a closed interval gets a little bit more involved so we say f is continuous over an open interval from A to B so the parenthesis instead of brackets this shows that we're not including the endpoints so this would be all of the points between x equals a and x equals B but not equally x equals a and x equals B so f is continuous over this open interval if and only if if and only if F is continuous F is continuous over every point in every over every point in the interval so let's do a couple of examples of that so let's say we're talking about the open interval from negative seven to negative five is f continuous over that interval let's see we're going from negative seven to negative five and there's a couple of ways you could do it there's a not so mathematically rigorous way where you could say hey look if I start here I can get all the way to negative five without having to pick up my pencil if you want to do more rigorously and you actually had the definition of the function you might be able to do a proof that for any of these points over the interval that the limit as X approaches any one of these points of f of X is equal to the value of the function at that point it's harder to do when you only have a graph when you only have a graph you can only just do it by inspection and say okay I can go from that point to that point without picking up my pencil so I feel pretty good about it now let's do another interval let's say these let me put check mark here that is continuous let's think about the interval from negative two to positive one the open interval so this is interesting because the function at negative two is up here and so if you really wanted to start negative two you would have to start here and then jump immediately down as soon as you get slightly larger than negative two and then keep going but this is an open interval so we're not actually concerned with what exactly happens at negative two we're concerned what happened when we are all the numbers larger than negative two so we would actually start right over here and then we would go to one and once again based on the intuitive I didn't have to pick up my pen idea this function would be continuous over this over this interval so what's an example of an interval where the function would not be continuous well think about the interval from well this is a pretty straightforward one the open interval from 3 to 5 the function is here when X is equal to but if we wanted to get to five it looks like we're asymptoting it looks like we're asymptoting up towards infinity and we just keep on going for a very long time and then we would have to pick up our pencil and jump over and then we would come back down right over here and so here we are not continuous over that interval so now let's think about the more the slightly more involved interval the slightly more involved case is when you have a closed interval f is continuous over the closed interval from A to B so this includes not just the points between a and B but eight 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 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 7 to negative 5 well this one is still reasonable you know 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 end point and at negative 7 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 clothes and you could have one side open one side closed but let's just do the closed interval from negative 3 to negative 2 so notice I did not have to pick up my pencil I'm including negative 3 and I'm getting all the way to negative 2 if you knew the analytic definition of dysfunction you could prove that hey the limit at any of these points inside between negative 3 and negative 2 is equal to the value of the function negative 3 the function is clearly at negative 3 the function is just plain old continuous the two-sided limit approaches the value of the function but at negative 2 the two-sided limit does not exist when you approach from the left looks like you're approaching zero f of X is equal to 0 when you approach from the right it looks like f of X is approaching negative 3 so even though the two-sided limit does not exist we can still be good because the left handed limit does exist and the left handed limit is approaching the value of the function so we actually are continuous over that interval but then if we did the interval if we did the closed interval from negative 2 to negative 2 to 1 pause the video and think about based on what we just talked about are we continuous over this interval well we're going from negative 2 to 1 and negative 2 is the lower bound so is this right over here is this right over here true is the limit as we approach negative 2 from the right is that the same thing as F of negative 2 well the limit as we approach from the right seems to be approaching negative 3 and F of negative 2 is 0 so this limit does not this these two things the limit as we approach from the right and the value of the function are not the same and so we we do not have that I guess you'd say that one-sided continuity at negative 2 and it also makes sense if I start at negative 2 let me just in a color you can see if I start at negative 2 and I want to go the rest of the interval to 1 I have to pick up my pencil pick up my pencil go here and the keep on going so this is we are not continuous over that interval