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Current time:0:00Total duration:11:14

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 that's the y-axis that is the x-axis and if I were to draw a function if I were to draw a function let's say f of X looks like something like this looks 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 well over here we see the function just jumps all of a sudden from this point to this point right over here this is not this is not continuous 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 continuity so you'd say this is not continuous it's obvious it's 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 so that 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 discontinuity T nua T 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 redefine the function so it didn't so it wasn't up here but it was right over here then the function was continuous so you can kind of remove 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'd say well look it looks all connected all the way there aren't any jumps over here no removable discontinuities over here this one looks continuous 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 epsilon 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 hope passes muster when we look at all of these things up here so let me draw an interval an interval right over here so it's between that x-value and that exercise 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 is not at the edge of my boundary so this is an interior point for my interval this would be an endpoint and this would also be an endpoint we'd say it's continuous at an interior point so continuous at interior point interior to my interval means that the limit the limit as let's say at interior point 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 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 this is if these would have somehow been able to pass for continuity 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 if this does not equal 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 this does not hold up in order for the limit to be equal to f of C the limit from both 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 reset it up so let me let me make sure that looks like a hole right over there so we see here that 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 pass our test the limit as X approaches this value is equal the limit the limit as we approach as X approaches that value is indeed 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 this is continuity for an interior point and let's think about continuity continuity on the right over here continuity continuity at boundary at boundary or let me call it endpoint actually that would be better at endpoint endpoints E and Point C so let's first consider it's the left endpoint if left endpoint so what am I talking about a left endpoint let me draw my let me draw my axes x-axis y-axis and let me draw my interval so let's say my this is the left endpoint of my interval this is the right endpoint of my interval and let me draw the function over that over that interval looks something looks something like this so when we talk about a left endpoint we're talking about RC RC being right over here it is the left endpoint so if we're talking about a left end point we are continuous at so we are continuous at C means or to say that we're continuous at this left endpoint C that means that the limit of f of X as X approaches C well we can't even approach C from the left hand side we have to approach from the right 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 end point where we would not be to set an endpoint well I can imagine a graph I can imagine a graph that looks something like this so here's our interval 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 looks something like that or there's no hole the function just it 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 not not continuous and you can imagine what do we do if we're dealing with the right endpoint so we say we're continuous at right endpoint and Point 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 so then 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 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 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 as we used limits to define a more rigorous definition of continuity