AP®︎/College Calculus AB
Continuity over an interval
A function ƒ is continuous over the open interval (a,b) if and only if it's continuous on every point in (a,b). ƒ is continuous over the closed interval [a,b] if and only if it's continuous on (a,b), the right-sided limit of ƒ at x=a is ƒ(a) and the left-sided limit of ƒ at x=b is ƒ(b).
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- A long time ago, when I was young, I was taught to designate "if and only if" with "iff." When did this double headed symbol which is not easy to draw displace the designation I learned in school?(15 votes)
- I use "iff" when explaining a logical biconditional in words. However, in logic and mathematics, when you are not actually explaining something but writing work out you use symbols. For example, using words, I could say "𝑄 if and only if 𝑃". Or, I could write this symbolically as:
𝑃 ⇔ 𝑄
It's similar to how I could write out: "𝑥 equals three" or simply "𝑥 = 3".
As such, I am sure that the double-sided arrow biconditional operator (the symbol itself) was probably developed after logicians first studied biconditionals themselves (as formal notation is developed far after the concepts themselves are developed). However, this still means a couple centuries ago (notation is pretty old by now).
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- Do people use (a,b] to describe an interval which involves all the points between a and b and the point a?(3 votes)
- (a, b] would correspond to a < x ≤ b. The interval would NOT include a.
- How can you prove that a function is continuous over every point in an interval? I get the concept--no picking up pencils--but I don't know how to prove it properly.(4 votes)
- You need to show that for every r > 0 there exists a d > 0 such that |x-y| < d => |f(x) - f(y)| < r.
Example, let f(x) = x. Then for any r > 0 pick r = d then |x - y < d = r => |f(x) - f(y)| = | x - y| < r.
Example, let f(x) = x^2. Then for any r > 0 observe that |x^2-y^2| = |x-y|*|x+y|. So if |x-y| < r/(1 + |x - y) then |x^2 - y^2| < r. (Note the use of 1/(1+ ..) so that if x = y I don't divide by zero).
If the function is differentiable, then it will be continuous on your open interval!(4 votes)
- For the graph that Sal gives, would the interval between (2, 4) or (2, 4] be continuous?
At least for the first interval (2, 4), technically you wouldn't have to "pick up your pencil" when drawing the graph. However, the limit as x approaches 4 doesn't exist, so it kind of contradicts itself when determining if the interval is continuous or not.(2 votes)
- The function is continuous on (2, 4), but not on (2, 4]. The limit as x goes to 4 doesn't exist, but that doesn't matter because 4 isn't in the interval (2, 4).(3 votes)
- Okay so there is a function f(x) = 1/x and we see that the function is not defined at x=0
but why does it make it discontinuous at that point?(2 votes)
- The way I understand it, that question depends on the domain under consideration. Since f(x) is not defined at x=0, it could be argued that the domain is (-∞,0) U (0,∞) in which case 0 is not part of the domain and the function is continuous over its domain. However, if you consider the domain to be all real numbers, it is not continuous. To be continuous at a point (say x=0), the limit as x approaches 0 must equal to the actual function evaluated at 0. The function f(x)=1/x is undefined at 0, since 1/0 is undefined. Therefore there is no way that the f(0) = lim x->0 f(x).(1 vote)
- how is x continuous at x=-2 when the function clearly doesn't follow all three steps of the continuity definition? The lim has to exist at a point for the function to be continuous at that point.(2 votes)
- Watch the video again closely. Sal took the value of the function over the open interval (-2,1), which does not include the jump in the value at -2.(1 vote)
- This video is VERY confusing with respect to closed intvl: in the practice I took, it emphasizes that EVERY pt including the end pts must be continuous, yet this video stipulates that continuity can be continuous at "a" coming from the Left "-", or "b" coming from the right "+" (see5:57in the 9 min video. It would seem to me that every pt makes more sense with respect to closed intvl.
By the way I too much prefer the designation iff for if and only if, but then at the ripe young age of 70, I took Calc 45 yrs ago.
- yeah I got the same issue on the quiz I was taking. But referring to the video it said as you said... very confused :((1 vote)
- What about the open interval (3,4) in the graph shown? Would it be continuous with the asymptote?(1 vote)
- Yes it would still be continuous because in that interval, 4 is excluded. However, as it approaches 4, the number will get extremely large, and only get larger and larger the closer you get to 4. If you tried to include 4 as part of the interval (3,4], then it is discontinuous at 4.(2 votes)
- It seems contradictory to say a closed interval is continuous when an endpoint of that interval is not continuous. For example, in the video, the closed interval [-3,-2] is considered continuous, but the -2 endpoint, i.e. point -2,0, is not continuous. I know this is by definition, but it tripped me up on the unit test as I made the mistake of assuming that the endpoint of a continuous closed interval was continuous. I won't make that mistake again ;-)(1 vote)
- Intervals are not continuous - continuity is a property of functions. What I think you're referring to is that closed intervals are connected, i.e. you can draw a straight line from any two pts in the closed interval.(2 votes)
- when does a limit gets defined(2 votes)
- A limit is defined if the function is continuous, and the value of the function is equal as the limit is approached from both sides.(0 votes)
- [Instructor] 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 gonna 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. If we approach from the right, we're getting this value. Well, in order for the function to be continuous, if I had 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. 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 with that out the way, let's discuss continuity over intervals. Let me delete this really fast, so I have space to work with. So we say, so I'm gonna first talk 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 parentheses instead of brackets, this shows that we're not including the endpoint. So this would be all of the points between x equals a and x equals b, but not equaling 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, 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 the 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 wanted 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 the, so let me put a 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 at 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 happens 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 three to five. The function is here when x is equal to three. 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 the endpoints as well, if and only if, f is continuous over the open interval and the one-sided limits. Let me right 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 seven to negative five, 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 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. I'm including negative three, and I'm getting all the way to negative two. If you knew the analytic definition of this function, you could prove that, hey, the limit at any of these points inside, between negative three and negative two, is equal to the value of the function. Negative three, the function is clearly, at negative three, the function is just plain old continuous. The two-sided limit approaches the value of the function. But at negative two, the two-sided limit does not exist. When you approach from the left, it looks like you're approaching zero. F of x is equal to zero. When you approach from the right, it looks like f of x is approaching negative three. 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 two to negative two to one, 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 two to one, and negative two is the lower bound. So is this right over here, is this right over here true? Is the limit as we approach negative two from the right, is that the same thing as f of negative two? Well, the limit as we approach from the right seems to be approaching negative three, and f of negative two is zero. So this limit does not, this, 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 do not have that, I guess you could say that one-sided (laughs) continuity at negative two. And that also makes sense. If I start at negative two, let me do this in a color you can see, if I start at negative two and I want to go the rest of the interval to one, I have to pick up my pencil. Pick up my pencil, go here, and then keep on going. So this is, we are not continuous over that interval.