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# Types of discontinuities

AP.CALC:
LIM‑2 (EU)
,
LIM‑2.A (LO)
,
LIM‑2.A.1 (EK)

## Video transcript

what we're going to do in this video is talk about the various types of discontinuities that you've probably seen when you took algebra or precalculus but then relate it to our understanding of both two sided limits and one-sided limits so let's first review the classification of discontinuities so here on the left you see that this curve looks just like y equals x squared until we get to x equals 3 and instead of it being 3 squared it does that at this point you have this opening and instead the function at 3 is defined at 4 but then it keeps going it looks just like y equals x squared this is known as a point or a removable discontinuity and it's called that for obvious reasons you're discontinuous at that point you might imagine of defining or redefining the function at that point so it is continuous so the discontinuity is removable but then how does this relate to our definition of continuity well let's remind ourselves our definition of continuity we say f is continuous continue us if and only if I let me write F continuous at x equals C if and only if the limit as X approaches C of f of X is equal to the actual value of the function when X is equal to C so why does this one fail well the two-sided limit actually exists you could find if we say C in this case is 3 the limit as X approaches 3 of f of X it looks like if you graphically inspect this and I actually know this is the graph of y equals x squared except at that discontinuity right over there this is equal to 9 but the issue is is the way this graph has been depicted this is not the same thing as the value of the function this function f of 3 the way it's been graphed F of 3 is equal to 4 so this is a situation where this two sided limit exists but it's not equal to the value that function you might see other circumstances where the function isn't even defined there so that isn't even there until once again the limit might exist but the function might not be defined there so in either case you aren't going to meet this criteria for continuity and so that's how a point or removable discontinuity why it is discontinuous with regards to our limit definition of continuity so now let's look at this a second example if we looked at our intuitive continuity test if we were just try to trace this thing we see that once we get to x equals two I have to pick up my pencil to keep tracing it and so that's a pretty good sign that we are discontinuous we see that over here as well if I'm tracing this function I got to pick up my pencil too I can't go through that point I have to jump down here and then keep going right over there so either case have to pick up my pencil and so intuitively it's discontinuous but this particular type of discontinuity where I'm making a jump from one point and I'm making a jump down here to continue it is intuitively called a jump discontinuity discontinuity and this is of course a point removable discontinuity and so how does this relate to limits well here the left and right handed limits exist but they're they're not the same thing so you don't have a two sided limit so for example for this one in particular but from for all the X values up to including x equals two this is the graph of y equals x squared and then for X greater than two is the graph of square root of X so in this scenario if you take the limit of f of X as X approaches two from the left from the left this is going to be equal to four you're approaching this value and that actually is the value of the function but if you were to take the limit as X approaches two from the right of f of X what is that going to be equal to or approaching from the right this is actually the square root of x so it's approaching the square root of two you can't you would know it's the square root of two just by looking at this I don't know that just because when I when I went on to decimals and define the function that's the function that I used but it's clear even visually that you're approaching two different values when you approach from the left then when you approach from the right so even though the one-sided limits exist they're not approaching the same thing so the two-sided limit doesn't exist and if the two-sided limit doesn't exist for sure cannot be equal to the value of the function there even if the function is defined so that's why did the jump discontinuity is failing this test and once again it's intuitive you're seeing that hey I got a jump I got to pick up my pencil these two things are not connected to each other finally what you see here is when you learned precalculus often known as an asymptotic discontinuity asymptotic asymptotic discontinuity discontinuity and intuitively you have an asymptote here if you it's a vertical asymptote at x equals two if I were to try to trace the graph from the left I'd be I would just keep on going in fact I would be doing it forever because it would be it'd be infinitely it'd be unbounded as I get closer and closer to x equals two from the left and if I get to try to get to x equals 2 from the right once again I get unbounded up but even if I could and when I say it's unbounded it goes to infinity so it's actually impossible and you know in a forum or in a mortal's life's band to try to trace the whole thing but you get the sense that hey there's no way that I could draw from here to here without picking up my pencil and if you want to relate it to our notion of limits it's that both the left and right handed limits are unbounded so they officially don't exist so if they don't exist then we can't meet this conditions so if I were to say the limit as X approaches 2 from the left hand side of f of X we can see that it goes unbounded in the negative direction you might sometimes see someone write something like this negative infinity but that's a little hand wavy with the math the more correct way to say it it's just unbounded unbounded and likewise if we thought about the limit as X approaches two from the right of f of X it is now unbounded towards positive infinity so this once again this is also this is also unbounded and because it's unbounded in this limit does not exist it can't meet these conditions and so we are going to be discontinuous so this is a point or removable discontinuity jump discontinuity I'm jumping and then we have these asymptotes of vertical asymptotes this is an asymptotic discontinuity
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