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Continuity introduction

Sal introduces a formal definition of continuity at a point using limits. Created by Sal Khan.

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  • duskpin ultimate style avatar for user Victor Pellen
    Okay, here's an odd case: What about 1/x? 1/x is not defined at 0, but the limit of 1/x as x -> 0 is ALSO not defined. Or, rather, it doesn't exist. Does this mean that 1/x qualifies as continuous, or are "function is not defined" and "limit does not exist" considered different things? My intuition says "1/x is not continuous" simply because, well, just look at it.

    Is there ANY function that's undefined at a given point that's still considered continuous, or must a function be defined at every point in order to be continuous?
    (31 votes)
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  • male robot hal style avatar for user Sergey Yegorov
    What is epsilon-delta rigorous definition of limits?
    (29 votes)
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  • aqualine sapling style avatar for user sim.harrypotter
    How many methods are there to know whether a function is continuous or not?
    (11 votes)
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    • blobby green style avatar for user GreenMaple75
      1) Use the definition of continuity based on limits as described in the video:
      The function f(x) is continuous on the closed interval [a,b] if:
      a) f(x) exists for all values in (a,b), and
      b) Two-sided limit of f(x) as x -> c equals f(c) for any c in open interval (a,b), and
      c) The right handed limit of f(x) as x -> a+ equals f(a) , and
      d) The left handed limit of f(x) as x -> b- equals f(b).
      2) Use the pencil test: a continuous function can be traced over its domain without lifting the pencil off the paper.
      3) A continuous function does not have gaps, jumps, or vertical asymptotes.
      4) Differentiability implies continuity.
      5) Classification of functions based on continuity. Examples:
      All polynomial functions are continuous over their domain.
      All rational functions are continuous except where the denominator is zero.
      The composition of two continuous functions is continuous.
      The inverse of a continuous function is continuous.
      Sine, cosine, and absolute value functions are continuous.
      Greatest integer function (f(x) = [x]) and f(x) = 1/x are not continuous.
      Sign function and sin(x)/x are not continuous over their entire domain.
      (26 votes)
  • piceratops ultimate style avatar for user Firedrake969
    Would f(x) = sqrt(x) be continuous? It starts at (0, 0) but its domain isn't all real numbers.
    (7 votes)
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  • piceratops ultimate style avatar for user Akshaj Jumde
    Can we call a function continuous if its limit at one point does not exist but the limit exists at the other point ?
    (4 votes)
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    • leaf grey style avatar for user Qeeko
      You should be more clear about what you mean by "(…) but the limit exists at the other point?".

      Recall that we call a function continuous on a given set if and only if it is continuous at every point of that set. As such, if things go bad at one point, it does not really help if things are well-behaved at some other point. There are three cases if the limit at a given point in the domain of the function fails to exist. They come down to the characterisation of the point in question.

      To be rather precise, suppose ƒ: A → R is a function defined on some non-empty subset A of R. Let a ∈ A be a point where the limit lim (x→a) ƒ(x) fails to exist.

      Case I: a is an isolated point of A. We call a an isolated point of A if and only if there exists some neighbourhood of a whose intersection with A \ {a} is empty. This is the most trivial case. If a is an isolated point of A, it is not meaningful to speak of the limit of ƒ at a (since it is undefined). Hence, the limit does not exist. In spite of this, ƒ is continuous at a.

      Case II: a is an interior point of A. This means that there is some neighbourhood of a entirely contained in A. Most texts on elementary calculus only define limits at such points. If the limit of ƒ does not exist at such a point a, then ƒ is not continuous at a, i.e., is discontinuous at a.

      Case III: a is a limit point of A, but not an interior point of A. We call a a limit point of A if and only if every neighbourhood of a contains some point of A different from a itself. Since a is not an interior point of A, the limit of ƒ at a does not exist (is undefined). In this case we can not assert any general conclusion that will hold for every such function ƒ and point a; ƒ might be continuous at a, or it might be discontinuous. If there exists a sequence {a(n)} of points in A such that a(n) ≠ a for every n and such that a(n) → a (that is, the sequence converges to a), but the sequence {ƒ[a(n)]} does not converge to ƒ(a), then ƒ is not continuous at a. If no such sequence exists, then ƒ is continuous at a.
      (5 votes)
  • leaf green style avatar for user Nicole
    What is the difference between a function f(x) being discontinuous, and a function f(x) not being differentiable?
    (5 votes)
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    • piceratops ultimate style avatar for user Just Keith
      A function is not differentiable if it contains a point that has one instantaneous slope if measured from the positive x direction and a different slope if measured from the negative x direction. This is true even if the function is continuous.

      You can sometimes restrict the domain and differentiate only a portion of the function that is differentiable, but the function as a whole would not be differentiable.
      (2 votes)
  • leaf orange style avatar for user Z.K.
    Is the function f(x)= sqrt(x) continuous? I understand that it is continuous for every value in its domain, but do you consider that it has no values for x<0?
    (2 votes)
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  • piceratops ultimate style avatar for user Simon
    If a function contains an absolute value of some sort, can it then be continuous over some interval even if it's not derivable all over that interval?
    (4 votes)
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  • mr pants teal style avatar for user bosa weluche-ume
    How is the Epsilon-Delta definition used or even needed for Calculus, both differentiation and integration, I know that they are used but I don't understand why or how, please and thank you
    (3 votes)
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  • aqualine ultimate style avatar for user Lochie.3.142
    So I'm guessing a graph is not defined as continuous if you have a situation where one side - or both - of a limit is approaching infinity (hence the limit does not exist, as has been discussed), but I was wondering what exactly this discontinuity is called, if it has a name.
    (3 votes)
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    • leaf green style avatar for user Hiya
      There are many kinds of discontinuity, some of them are:
      1) Missing point - Picture a smooth curve with just one "hole" in it, that is the function is not defined at that point
      2) Isolated point - Similar to case 1), except the function is defined at that point, just not in the line of the curve
      3) Jump - Eg: The function consists of two smooth curves which aren't connected, like the first example Sal shows
      4) Infinte - When the graph goes to positive or negative infinity and comes back from positive or negative infinity, think tanx graph
      Hope this helped!
      (2 votes)

Video transcript

What I want to do in this video is talk about continuity. And continuity of a function is something that is pretty easy to recognize when you see it. But we'll also talk about how we can more rigorously define it. So when I talk about it being pretty easy to recognize, let me draw some functions here. So let's say this is the y-axis, that is the x-axis. And if I were to draw a function, let's say f of x looks like something like this. And I would say over the interval that I've drawn it, so it looks like from x is equal to 0, to maybe that point right over there, is this function continuous? Well you'd say no, it isn't. Look, we're over here, we see the function just jumps all of a sudden, from this point to this point right over here. This is not continuous. And you might even say we have a discontinuity at this value of x, right over here. We would call this a discontinuity. And actually this type of discontinuity is called a jump discontinuity. So you would say this is not continuous. It's obvious that these two things do not connect. They don't touch each other. Similarly, if you were to look at a function that looked like-- let me draw another one-- y and x. And let's say the function looks something like this. Maybe right over here, looks like this, and then the function is defined to be this point right over there. Is the function continuous over the interval that I've depicted right over here? And you would immediately say no, it isn't. Because right over at this point, the function goes up to this point, just like this. And this kind of discontinuity-- this is the discontinuity-- is called a removable discontinuity. Removable. One could make a reasonable argument that this also looks like a jump, but this is typically categorized as a removable discontinuity. Because if you just re-define the function so it wasn't up here, but it was right over here, then the function is continuous. So you can kind of remove the discontinuity. And then finally, if I were to draw another function-- so let me draw another one right over here-- x, y. And ask you, is this one continuous over the interval that I've depicted? And you would say, well, look. Yeah, it looks all connected all the way. There aren't any jumps over here. No removable discontinuities over here. This one looks continuous. And you would be right. So that's the general sense of continuity. And you can kind of spot it when you see it. But let's think about a more rigorous definition of one. And since we already have a rigorous definition of limits, the epsilon delta definition, gives us a rigorous definition for limits. It's a definition for limits. So we can prove when a limit exists, and what the value of that limit is. Let's use that to create a rigorous definition of continuity. So let's think about a function over some type of an interval. So let's say that we have-- so let me draw another function. Let me draw some type of a function. And then we'll see whether our more rigorous definition of continuity passes muster, when we look at all of these things up here. So let me draw an interval right over here. So it's between that x value and that x value. This is the x-axis, this is the y-axis. And let me draw my function over that interval. Over that interval it looks something like this. So we say that a function is continuous at an interior point. So an interior point is a point that's not at the edge of my boundary. So this is an interior point for my interval. This would be an end point, and this would also be an end point. We'd say it's continuous at an interior point. So continuous at interior point, interior to my interval, means that the limit as, let's say at interior point c, so this is the point x is equal to c. We can say that it's continuous at the interior point c if the limit of our function-- this is our function right over here-- if the limit of our function as x approaches c is equal to the value of our function. Now does this make sense? Well, what we're saying is, is at that point, well this is f of c right over there. And the limit as we approach that is the same thing as the value of the function. Which makes a lot of sense. Now let's think about it. If these would have somehow been able to pass for continuous in that context. Well, over here, let's say that this is our point c. f of c is right over there. That is f of c. Now is it the case that the limit of f of x, as x approaches c, is equal to f of c? Well, if we take the limit of f of x as x approaches c from the positive direction, it does look like it is f of c. It does look like it's equal to f of c. But if we take the limit-- but this does not equal the limit of f of x as x approaches c from the negative direction. As we go from the negative direction, we're not approaching f of c. So therefore, this does not hold up. In order for the limit to be equal to f of c, the limit from both the directions needs to be equal to it. And this is not the case. So this would not pass muster by our formal definition, which is good. Because we see visually this one is not continuous. What about this one right over here? And let me re-set it up. So let me make sure that that looks like a hole right over there. So we see here, what is the limit? The limit-- and this is our c, right over here-- the limit of f of x as x approaches c, let's say that that is equal to L. And so that, we've seen many limits like this before, that's L right over there. And it's pretty clear just looking at this is that L does not equal f of c. This right over here is f of c. So once again, this would not pass our test. The limit of f of x as x approaches c, which is this right over here, is not equal to f of c. So once again, this would not pass our test. And here, any of the interior points would pass our test. The limit as x approaches this value is equal to the function evaluated at that point. So it seems to be good for all of those. Now let's give a definition for when we're talking about boundary points. So this is continuity for an interior point. And let's think about continuity at boundary-- or let me call it endpoint, actually, that would be better-- at endpoint c. So let's first consider just the left endpoint. If left endpoint-- so what I'm talking about, a left endpoint? Let me draw my axes, x-axis, y-axis. And let me draw my interval. So let's say this is the left endpoint of my interval, this is the right endpoint of my interval. And let me draw the function over that interval. Looks something like this. So when we talk about a left endpoint, we're talking our c being right over here. It is the left endpoint. So if we're talking about a left endpoint, we are continuous at c means-- or to say that we're continuous at this left endpoint c-- that means that the limit f of x as x approaches c, well, we can't even approach see from the left hand side. We have to approach from the right. Is equal to f of c. And so this is really kind of a, we can only approach something from one direction. So we can't just say the limit in general, but we can say the limit from one side. So it's really very similar to what we just said for an interior point. And we see over here it is indeed the case, as x approaches c, our function is approaching this point right over here. Which is the exact same thing as f of c. So we are continuous at that point. What's an example where an endpoint-- where we would not be continuous and an end point? Well, I can imagine a graph that looks something like this. So here's our interval, and maybe our function. So at c it looks like that. There's a little hole right there. And then it would look something like that. Or there's no hole, the function just has a removable discontinuity right over there. At least visually it looks like that. And you see that this would not pass the test. Because the limit as we approach c from the positive direction is right over here. That's the limit. But f of c is up here. So f of c does not equal the limit as x approaches c from the positive direction. So this would not be continuous. And you could imagine, what do we do if we're dealing with the right endpoint? So we say we're continuous at right endpoint c if-- so let me draw that, do my best attempt to draw it-- so this is my x-axis, this is my y-axis. Let me draw my interval that I care about, say it looks something like this. A right endpoint means c is right over there. And we can say that we are continuous at x, the function is continuous at x equals c means that the limit of f of x as x approaches c-- now we can't approach it from both sides. We can only approach it from the left hand side. As x approaches c from the negative direction, is equal to f of c. If we can say that, if this is true, then this implies that we are continuous at that right endpoint, c, and vice versa. And a situation where we're not? Well you could imagine instead of this being defined right at that point, you could create, you could say the function jumps up. Just like we did right over there. So once again, continuity, not a really hard to fathom idea. Whenever you see the function just all of a sudden jumping, or there's kind of a gap in it, it's a pretty good sense that the function is not connected there. It's not continuous. But what we did in this video is, we used limits to define a more rigorous definition of continuity.