Proof: Differentiability implies continuity

- [Voiceover] What I hope to do in this video is prove that if a function is differentiable at some point, C, that it's also going to be continuous at that point C. But, before we do the proof, let's just remind ourselves what differentiability means and what continuity means. So, first, differentiability. Differentiability So, let's think about that, first. And it's always helpful to draw ourselves a function. So, that's our Y-axis. This is our X-axis. And let's just draw some function, here. So, let's say my function looks like this and we care about the point X equals C, which is right over here. So, that's the point X equals C, and then, this value, of course, is going to be F of C. F of C. And one way that we can find the derivative at X equals C, or the slope of the tangent line at X equals C is, we could start with some other point. Say, some arbitrary X out here. So, let's say this is some arbitrary X out here. So, then, this point right over there, this value, this Y value, would be F of X. Would be F of X. This graph, of course, is a graph of Y equals F of X. And we can think about finding the slope of this line, this secant line between these two points, but then, we can find the limit as X approaches C. And as X approaches C, this secant, the slope of the secant line is going to approach the slope of the tangent line, or, it's going to be the derivative. And so, we could take the limit... The limit as X approaches C, as X approaches C, of the slope of this secant line. So, what's the slope? Well, it's gonna be change in Y over change in X. The change in Y is F of X minus F of C, that's our change in Y right over here. This is all review, this is just one definition of the derivative, or one way to think about the derivative. So, it's going to be F of X minus F of C, that's our change in Y, over our change in X. Over our change in X, which is X minus C. It is X minus, X minus C. So, if this limit exists, then, we're able to find the slope of the tangent line at this point, and we call that slope of the tangent line, we call that the derivative at X equals C. We say that this is going to be equal to F prime, F prime of C. All of this is review. So, if we're saying, one way to think about it, if we're saying that the function, F, is differentiable at X equals C, we're really just saying that this limit right over here actually exists. And if this limit actually exists, we just call that value F prime of C. So, that's just a review of differentiability. Now, let's give ourselves a review of continuity. Continuity. So, the definition for continuity is if the limit as X approaches C of F of X is equal to F of C. Now, this might seem a little bit, you know, well, it might pop out to you as being intuitive or it might seem a little, well, where did this come from, well, let's visualize it and hopefully it'll make some intuitive sense. So, if you have a function, so, let's actually look at some cases where you're not continuous. And that actually might make it a little bit more clear. So, if you had a point discontinuity at X equals C, so, this is X equals C, so, if you had a point discontinuity, so, lemme draw it like this, actually. So, you have a gap, here, and X equals, when X equals C, F of C is actually way up here. So, this is F of C, and then, the function continues like this. The limit, as X approaches C of F of X is going to be this value, which is clearly different than F of C. This value right over here, if you take the limit, if you take the limit as X approaches C of F of X, you're approaching this value. This, right over here, is the limit, as X approaches C of F of X, which is different than F of C. So, it makes it, so, this definition of continuity seems to be good, at least for this case, because this is not a continuous function, you have a point discontinuity. So, for at least in this case, our, this definition of continuity would properly identify this as not a continuous function. Now, you could also think about a jump discontinuity. You can also think about a jump discontinuity. So, let's look at this. And all this is, hopefully, a little bit of review. So, a jump discontinuity at C, at X equals C, might look like this. Might look like this. So, this is at X equals C. So, this is X equals C right over here. This would be F of C. But, if you tried to find a value at the limit as X approaches C of F of X, you'd get a different value as you approach C from the negative side, you would approach this value, and as you approach C from the positive side, you would approach F of C, and so, the limit wouldn't exist. So, this limit right over here wouldn't exist in the case of jump, of this type of a jump discontinuity. So, once again, this definition would properly say that this is not, this one right over here, is not continuous, this limit actually would not even exist. And then, you could even look at a, you could look at a function that is truly continuous. If you look at a function that is truly continuous. So, something like this. Something like this. That is X equals C. Well, this is F of C. This is F of C. And if you were to take the limit as X approaches C, as X approaches C from either side of F of X, you're going to approach F of C. So, here, you have the limit as X approaches C of F of X, indeed, is equal to F of C. So, it's what you would expect for a continuous function. So, now that we've done that review of differentiability and continuity, let's prove that differentiability actually implies continuity, and I think it's important to kinda do this review, just so that you can really visualize things. So, differentiability implies this limit right over here exists. So, let's start with a slightly different limit. Lemme draw a line, here, actually. Lemme draw a line just so we're doing something different. So, let's take, let us take the limit as X approaches C of F of X, of F of X minus F of C. Of F of X minus F of C. Well, can we rewrite this? Well, we could rewrite this as the limit, as X approaches C, and we could essentially take this expression and multiply and divide it by X minus C. So, let's multiply it times X minus C. X minus C, and divide it by X minus C. So, we have F of X minus F of C, all of that over X minus C. So, all I did is I multiplied and I divided by X minus C. Well, what's this limit going to be equal to? This is going to be equal to, it's going to be the limit, and I'm just applying the property of limit, applying a property of limits, here. So, the limit of the product is equal to the same thing as a product of the limits. So, it's the limit as X approaches C of X minus C, times the limit, lemme write this way, times the limit as X approaches C of F of X minus F of C, all of that over X minus C. Now, what is this thing right over here? Well, if we assume that F is differentiable at C, and we're going to do that, actually, I should have started off there. Let's assume 'cause we wanted to show the differentiability, it proves continuity. If we assume F differentiable, differentiable at C, well then, this right over here is just going to be F prime of C. This right over here, we just saw it right over here, that's this exact same thing. This is F prime, F prime of C. And what is this thing right over here? The limit as X approaches C of X minus C? Well, that's just gonna be zero. As X approaches C, there's gonna become, approach C minus C, that's just going to be zero. So, what's zero times F prime of C? Well, F prime of C is just going to be some value, so, zero times anything is just going to be zero. So, I did all that work to get a zero. Now, why is this interesting? Well, we just said, we just assumed that if F is differentiable at C, and we evaluate this limit, we get zero. So, if we assume F is differentiable at C, we can write, we can write the limit, I'm just rewriting it, the limit as X approaches C of F of X minus F of C, and I could even put parenthesis around it like that, which I already did up here, is equal to zero. Well, this is the same thing, I could use limit properties again, this is the same thing as saying, and I'll do it over, well, actually, lemme do it down here. The limit as X approaches C of F of X minus the limit as X approaches C of F of C, of F of C, is equal to zero. The different, the limit of the difference is the same thing as the difference of the limits. Well, what's this thing over here going to be? Well, F of C is just a number, it's not a function of X anymore, it's just, F of C is going to valuate it to something. So, this is just going to be F of C. This is just going to be F of C. So, the limit of F of X as X approaches C, minus F of C is equal to zero. Well, just add F of C to both sides and what do you get? Well, you get the limit as X approaches C of F of X is equal to F of C. And this is the definition of continuity. The limit of my function as X approaches C is equal to the function, is equal to the value of the function at C. This is, this means that our function is continuous. Continuous at C. So, just a reminder, we started assuming F differentiable at C, we use that fact to evaluate this limit right over here, which, we got to be equal to zero, and if that limit is equal to zero, then, it just follows, just doing a little bit of algebra and using properties of limits, that the limit as X approaches C of F of X is equal to F of C, and that's our definition of being continuous. Continuous at the point C. So, hopefully, that satisfies you. If we know that the derivative exists at a point, if it's differentiable at a point C, that means it's also continuous at that point C. The function is also continuous at that point.