Connecting differentiability and continuity: determining when derivatives do and do not exist
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Differentiability at a point: algebraic (function isn't differentiable)
- [Voiceover] Is the function given below continuous slash differentiable at x equals one? And they define the function g piece wise right over here, and then they give us a bunch of choices. Continuous but not differentiable. Differentiable but not continuous. Both continuous and differentiable. Neither continuous nor differentiable. And, like always, pause this video and see if you could figure this out. So let's do step by step. So first let's think about continuity. So for continuity, for g to be continuous at x equals one that means that g of one, that means g of one must be equal to the limit as x approaches one of g of g of x. Well g of one, what is that going to be? G of one we're going to fall into this case. One minus one squared is going to be zero. So if we can show that the limit of g of x as x approaches one is the same as g of one is equal to zero than we know we're continuous there. Well let's do the left and right handed limits here. So if we do the left handed limit, limit, and that's especially useful 'cause we're in these different clauses here as we approach from the left and the right hand side. So as x approaches one from the left hand side of g of x. Well we're going to be falling into this situation here as we approach from the left as x is less than one. So this is going to be the same thing as that. That's what g of x is equal to when we are less than one as we're approaching from the left. Well this thing is defined, and it's continuous for all real numbers. So we could just substitute one in for x, and we get this is equal to zero. So so far so good, let's do one more of these. Let's approach from the right hand side. As x approaches one from the right hand side of g of x. Well now we're falling into this case so g of x if we're to the right of one if values are greater or equal to one it's gonna be x minus one squared. Well once again x minus one squared that is defined for all real numbers. It's continuous for all real numbers, so we could just pop that one in there. You get one minus one squared. Well that's just zero again, so the left hand limit, the right hand limit are both equal zero, which means that the limit of g of x as x approaches one is equal to zero. Which is the same thing as g of one, so we are good with continuity. So we can rule out all of the ones that are saying that it's not continuous. So we can rule out that one, and we can rule out that one right over there. So now let's think about whether it is differentiable. So differentiability. So differentiability, I'll write differentiability, ability. Did I, let's see, that's a long word. Differentiability, alright. Differentiability, what needs to be true here? Well we have to have a defined limit as x approaches one for f of x minus f of one over, oh let me be careful, it's not f it's g. So we need to have a defined limit for g of x minus g of one over x minus one. And so let's just try to evaluate this limit from the left and right hand sides, and we can simplify it. We already know that g of one is zero. So that's just going to be zero. So we just need to find the limit as x approaches one of g of x over x minus one or see if we can find the limit. So let's first think about the limit as we approach from the left hand side of g of x over x minus. G of x over x minus one. Well as we approach from the left hand side, g of x is that right over there. So we could write this. Instead of writing g of x, we could write this as x minus one. X minus one over x minus one, and as long as we aren't equal to one, this thing is going to be equal as long as x does not equal one. X minus one over x minus one is just going to be one. So this limit is going to be one. So that one worked out. Now let's think about the limit as x approaches one from the right hand side of, once again, I could write g of x of g of one, but g of one is just zero, so I'll just write g of x over x minus one. Well what's g of x now? Well it's x minus one squared. So instead of writing g of x, I could write this as x minus one squared over x minus one, and so as long as x does not equal one, we're just doing the limit. We're saying as we approach one from the right hand side. Well, this expression right over here you have x minus one squared divided by x minus one, well, that's just going to give us x minus one. X minus one squared divided by x minus one is just going to be x minus one, and this limit, well this expression right over here is going to be continuous and defined for sure all x's that are not equaling one. Actually, let me, let me, well, it was before it was this, x minus one squared over x minus one. This thing over here, as I said, is not defined for x equals one, but it is defined for anything for x does not equal one, and we're just approaching one. And, if we wanted to simplify this expression, it would get, this would just be I think I just did this, but I'm making sure I'm doing it right. This is going to be the same this as that for x not being equal to one. Well this is just going to be zero. We could just evaluate when x is equal to one here. This is going to be equal to zero. And so notice, you get a different limit for this definition of the derivative as we approach from the left hand side or the right hand side, and that makes sense. This graph is gonna look something like, we have a slope of one, so it's gonna look something like this. And then right when x is equal to one and the value of our function is zero it looks something like this, it looks something like this. And so the graph is continuous the graph for sure is continuous, but our slope coming into that point is one, and our slope right when we leave that point is zero. So it is not differentiable over there. So it is continuous, continuous, but not differentiable.
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