![]() So now let's think about the more, the slightly more involved interval. And so here we are notĬontinuous over that interval. Up our pencil and jump over, and then we would comeīack down right over here. But if we wanted to get to five, it looks like we're asymptoting, it looks like we'reĪsymptoting up towards infinity and we just keep on goingįor a very long time. Straightforward one, the open interval from three to five. Would not be continuous? Well, think about the interval from, well, this is a pretty So what's an example of an interval where the function 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. Start right over here, and then we would go to one. But this is an open interval, so we're not actually concerned with what exactly happens at negative two, we're concerned what happens when we are all the numbers larger than negative two. Larger than negative two and then keep going. To start at negative two, you would have to start hereĪnd then jump immediately down as soon as you get slightly So this is interestingīecause the function at negative two is up here. Interval from negative two to positive one, the open interval. Put a check mark here, that is continuous. That point to that point without picking up my pencil, so I feel pretty good about it. When you only have a graph, you can only just do it by inspection, and say, okay, I can go from Points over the interval, that the limit as x approachesĪny one of these points of f of x is equal to If you wanted to do more rigorously and you actually had theĭefinition of the function, you might be able to do a proof, that for any of these Look, if I start here, I can get all the way to negative five without having to pick up my pencil. Not-so-mathematically-rigorous way, where you could say, hey, Is f continuous over that interval? Let's see, we're going from negative seven to negative five, and there's a couple of So let's say we're talkingĪbout the open interval from negative seven to negative five. ![]() ![]() So let's do a couple of examples of that. ![]() This open interval, if and only if, if and only if, f is continuous, f is continuous over every point in, over every point in the interval. Points between x equals a and x equals b, but not equaling xĮquals a and x equals b. So the parentheses instead of brackets, this shows that we're not So we say f is continuous over an open interval from a to b. Talk about an open interval, and then we're gonna talkĪbout a closed interval because a closed interval getsĪ little bit more involved. Let me delete this really fast, so I have space to work with. So with that out the way, let's discuss continuity over intervals. To pick up your pencil, this notion of connectedness, that you don't have any jumps or any discontinuities of any kind. Rigorous way of describing this notion of not having Without picking up my pen, well, the value of theįunction at that point should be the same as the limit. Well, in order for theįunction to be continuous, if I had to draw this function So if we approach, if we approach from the left, we're getting to this value. 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. And when we first introduced this, we said, hey, this looksĪ little bit technical, but it's actually pretty intuitive. Two-way arrows right over here, the limit of f of x as x approaches c is equal to f of c. So we say that f is continuous when x is equal to c, if and only if, so I'm gonna make these But to do that, let's refresh our memory about continuity at a point. Going to do in this video is explore continuity over an interval.
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