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# Continuity at a point

AP.CALC:
LIM‑2 (EU)
,
LIM‑2.A (LO)
,
LIM‑2.A.2 (EK)

## Video transcript

what we're going to do in this video is come up with a more rigorous definition for continuity and the general idea of continuity we've got an intuitive idea of the past is that a function is continuous at a point is if you can draw the graph of that function at that point without picking up your pencil so what do we mean by that and this is a very this what I just said is not that rigorous or not rigorous at all is that well let's think about the point right over here let's say that's our C if I can draw the graph at that point the value of the function at the point without picking up my pencil or my pen that it's continuous there so I could just start here and I don't have to pick up my pencil and there you go I can draw I can go through that point so we could say that our function is continuous there but if I had a function that looked somewhat different than that if I had a function that looked like this let's say that it is defined up until then and then there's a bit of a jump and then it goes like this well this would be very hard to draw with at this would be this function would be very hard to draw going through x equals c without picking up my pen let's see my pen is touching the screen touching the screen touching the screen how do I keep drawing this function without picking up my pen I would have to pick it up and then move back down here and so that is an intuitive sense that we are not continuous in this case right over here but let's actually come up with a formal definition for continuity and then see if it feels intuitive for us so the formal definition of continuity let's start here we'll start with continuity at a point so we could say the function f is continuous continuous at x equals C if and only if I draw this two-way arrow so if and only if the two-sided limit of f of X as X approaches C is equal to f of C so this seems very technical but let's just think about what it's saying it's saying look if the limit as we approach C from the left and the right of f of X if that's actually the value of our function there then we are continuous at that point so let's look at three examples let's look at one example we are we're by our picking up the pencil idea it feels like we are continuous at a point and then let's think about a couple of examples where it doesn't seem like we're continuous at a point and see how this more rigorous definition applies so let's say that my function so let's say this right over here is y is equal to f of X and we care about the behavior right over here when X is equal to C this is my x-axis that's my y-axis so we care about the behavior when X is equal to C and so notice from our first intuitive sense I can definitely draw this function as we go through x equals C without picking up my pencil so it feels continuous there there's no jumps or discontinuities that we can tell it just kind of keeps on going it seems all connected is one way to think about it but let's think about this definition well the limit as X approaches C from the left it is as we approach from the left it looks like it is approaching it looks like it is approaching f of C so this is the value F of f of C right over here and as we approach from the right as we approach from the right it also looks like it's approaching F of C and we are defined right at x equals C and it is the value that we are approaching from both the left or the right so this seems good in this scenario so now let's look at some scenarios that we would have to pick up the pencil as we draw the function through that point through that through that when that when X is equal to C so let's look at a scenario let's look at a scenario where we have what's often called a point discontinuity although you don't have to know at this point intended the formal terminology for it so let's say we have a function that let's see this is C and let's say our function looks something something like this so we go like this and at C let's say it's equal to that so f of C is right over here F of C would be that value but what's the limit as X approaches C so the limit as X approaches C and this would be a two-sided limit of f of X well this is as we approach from the left it looks like we are approaching this value right over here and from the right it looks like we are approaching that same value and so we could call that L and L is different than f of C and so in this case by our formal definition we will not be continuous at for F will not be continuous for X is equal at the point X or when X is equal to C and you can see that there if we tried to draw this okay my pencil is touching the paper touching the paper touching the paper oh if I needed to keep drawing this function I'd have to pick up my pencil move it over here then pick it up again and then jump right back down and but this rigorous definition is giving us the same conclusion the limit as we approach X equal C from the left and the right it's a different value than f of C and so this is not continuous not not continuous and let's think about another scenario let's think about a scenario and actually maybe you'll see one scenario where the limit the two sided limit doesn't even exist so there my AXYZ x and y and let's say it's doing something like this let's it's doing something like this and then it does something like this and goes like that and let's say that this right over here is our C and so let's see this this is f of C right over here that is we've got a little bit neater that is FOC and it does look like the limit as X approaches C from the left so from values less than C it does look like that is approaching F of C but if we look at the limit as X approaches C from the right that looks like it's approaching some other value that looks like it's approaching this value right over here let's call it al that's approaching L and L does not equal f of C and so in this situation the two-sided limit doesn't even exist we're approaching two different values when we approach from the left and from the right and since so the limit doesn't even exist at C this is definitely not going to be continuous and this matches up to our expectations with our little do we have to pick up the pencil test if I have to draw this I can leave my pencil it's on the paper it's on the paper it's on the paper it's on the paper how am I going to continue to draw this function this graph of the function without picking up my pencil pick it up put it back down and then keep drawing it so once again this right over here is not continuous both intuitively by our pick up the pencil definition and also by this more rigorous definition where in this case the limit the two-sided limit at x equals C doesn't even exist so we're definitely not be continuous but even when the two-sided limit does exist but the limit is a different value than the value of the function that will also not be continuous the only situation that is going to be continuous is if the two-sided limit approaches the same value as the value of the function and if that's true then we're continuous if we're continuous that is going to be true
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