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# Differentiability and continuity

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

## Video transcript

what we're going to do in this video is explore the notion of differentiability at a point and that is just a fancy way of saying does the function have a defined derivative at a point so let's just remind ourselves a definition of a derivative and there's multiple ways of writing this for the sake of this video I'll write it as the derivative of our function at Point C this is Lagrangian with this F prime the derivative of our function f at C is going to be equal to the limit as X approaches C of f of X minus f of C over X minus C and at first when you see this formula we've seen it before it looks a little bit strange but all it is is it's calculating the slope this is our change in the value of our function or you could think of it as our change in Y if Y is equal to f of X and this is our change in X and we're just trying to see well what what is that slope as X gets closer and closer to C as our change in X gets closer and closer to zero and we talk about that in other videos so I'm now going to make a few claims in this video and I'm not going to prove them rigorously there's another video that will go a little bit more into the proof direction but this is more to get an intuition and so the first claim that I'm going to make is if f is differentiable at x equals C at x equals C then F is continuous at x equals C so I'm saying if we know it's differentiable if we can find this limit if we can find this derivative at x equals C then our function is also continuous at x equals C it doesn't necessarily mean the other way around and actually we'll look at a case where it's not necessarily the case the other way around that if you're continuous then you're definitely differentiable but another way to interpret what I just wrote down is if you are not continuous then you definitely will not be differentiable if s not continuous at x equals C then f is not differentiable differentiable at X is equal to C so let me give a few examples of a non continuous function and then think about would we be able to find this limit so the first is where you have a discontinuity our function is defined at C it's equal to this value but you can see as X becomes larger than C it just jumps down and shifts right over here so what would happen if you are trying to find this limit well remember all this is is the slope of a line between when X is some arbitrary value let's say it's out here so that would be X this would be the point X comma f of X and then this is the point C comma F of C right over here so this is C comma F of C so if you find the left-sided limit right over here you're essentially saying okay let's find this slope and then let me get a little bit closer and let me put let's get X a little bit closer and then let's find the slope and then let's get X even closer than that and find this slope and in all of those cases it would be 0 the slope is 0 so one way to think about it the derivative or this limit as we approach from the left seems to be approaching zero but what about if we were take X's to the right so instead of our exes being there what if we were to take x's right over here well for this point X comma f of X our slope if we take f of X minus f of C over X minus C that would be the slope of this line if we get X to be even closer let's say right over here then this would be the slope of this line if we get even closer then this expression would be the slope of this line and so as we get closer and closer to X being equal to C we see that our slope is actually approaching negative infinity and most importantly it's approaching a very different value from the right this expression is approaching a very different value from the right as it is from the left and so in this case this this limit up here won't exist so we can clearly say this is not differentiable so once again not a proof here I'm just getting an intuition for if something isn't continuous it's pretty clear at least in this case that it's not going to be differentiable let's look at another case let's look at a case where we have what sometimes called a removable discontinuity or a point discontinuity so once again let's say we're approaching from the left this is X this is the point X comma f of X now what's interesting is where as this expression is the slope of the line connecting X comma f of X and C comma f of C which is this point not that point remember we have this removable discontinuity right over here and so this would be this expression is calculating the slope of that line and then if X gets even closer to C well then we're going to be calculating the slope of that line if X gets even closer to C we're going to be calculating the slope of that line and so as we approach from the left as X approaches C from the left we actually have a situation where this expression right over here is going to approach negative infinity and if we approach from the right if we approach with X is larger than C well this is our X comma f of X so we have a positive slope and then as we get closer it gets more positive more positive approaches positive infinity but either way it's not approaching a finite value and one side is approaching positive infinity or the other side's approaching negative infinity this the limit of this expression is not going to exist so once again I'm not doing a rigorous proof here but try to construct a discontinuous function where you will be able to find this it is very very hard and you might say well what about the situations where s does not even defined at C which for sure you're not going to be continuous if f is not defined at C well if F is not defined at C then this part of the expression wouldn't even make sense so you definitely wouldn't be differentiable but now let's ask another thing I've just given you good arguments for when you're not continuous you're not going to be differentiable but can we make us another claim that if you are continuous then you definitely will be differentiable well it turns out that there are for sure many functions an infinite number of functions that can be continuous at sea but not differentiable so for example this could be an absolute value function it doesn't have to be an absolute value function but this could be Y is equal to the absolute value of X minus C and why is this one not differentiable at C well think about what's happening think about this expression remember this expression all it's doing is calculating the slope between the point X comma f of X and the point C comma f of C so if X is say out here this is X comma f of X it's going to be calculate and as we're so as we take the limit for as X approaches C from the left we'll be looking at this slope and then as we get closer we'll be looking at this slope which is actually going to be the same in this case it would be a negative 1 so as X approaches C from the left this expression would be negative 1 but as we as X approaches C from the right this expression is going to be 1 the slope of the line that connects these points is 1 the slope on the line that connects these points is 1 so our the limit of this expression or I would say the value of this expression is approaching two different values as X approaches C from the left to the right from the left it's approaching negative 1 or it has constantly negative 1 and until it's approaching negative 1 you could say and from the right it's 1 and it's approaching 1 the entire time and so we know if you're approaching two different values from on the left-sided or the right-sided limit then this limit will not exist so here this is not not differentiable and even intuitively we think of the derivative as the slope of the tangent line and you could actually draw an infinite number of tangent lines here's one way to think about it you could say well maybe this is the tangent line right over there but why can't I make something like this the tangent line that only intersects at the point C comma zero and then you could keep doing things like that why can't that be the tangent line and you could go on and on and on so the big takeaways here at least intuitively and then in a future video I'm going to prove to you that if F is differentiable at C then it is continuous at C which can also be interpreted in that if you're not continuous at C then you're not going to be differentiable these two examples will hopefully give you some intuition for that but it's not the case that if something is continuous that it has to be differentiable it oftentimes will be differentiable but it doesn't have to be differentiable and this absolute value function is an example of a continuous function at C but it is not differentiable at C
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