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the result that I hope to show you or give you an intuition for in this video is something that we will use in a in the proof of the chain rule or an a proof of the chain rule actually mean we may do more than one proof of the chain rule but the result we're gonna look at is if we have some function U which is a function of X and we know that it is continuous it is continuous at x equals C so if we know this then that's going to imply that the change in U goes to zero as our change in X in this region around C goes to zero this is what I want to get an intuition for that a few is continuous at C then as our change in X around C gets smaller or it gets smaller and smaller suarez approaches zero then our change in U approaches zero as well so let's just to think about this and already even kind of prove it to ourselves a little bit more rigorously let's think about what it means to be continuous at X equal C well the definition of continuity is so this literally is the same thing as saying that the limit as X approaches C of U of X is equal to U of C that the the limit the limit that our function approaches is X approaches C is equal to the value of the function at C we don't have a a point discontinuity or a jump discontinuity if we had a jump discontinuity then then the limit wouldn't exist and we've seen that in previous videos now I'm just gonna manipulate this algebraically so it essentially gives us this conclusion right over here so this we can rewrite it's important to realize that U of C this is just going to be some value it looks like I've kind of you know maybe this is a function of X or something but no this is this is just going to be some value I've inputted C here and I've evaluated the function that and so there's gonna be some number it could be 5 or 7 or PI or negative 1 it's but it's just going to be some value some constant so I can treat it like a constant so this is going to be the same thing as saying the limit as X approaches C of U of X minus U of C is equal to is equal to zero and actually in the video where we prove that differentiability implies continuity we started with this and we and we proved that right over there we show that these two are equivalent things but hopefully you can even think about the intuition this is just if the limit is U of X approaches the limit of U of X as X approaches C is equal to this then when you evaluate this limit the limit as X approaches C well this thing is going to approach U of C because we saw it right up here U of C minus U of C is indeed going to be equal to zero so hopefully you don't feel like it's too much of a stretch and you can just subtract U of C from both sides and apply properties of limits and you can get this result as well but this is interesting because this essentially can take us to this that the idea that as our change in X gets smaller and smaller and smaller as it approaches zero then our change in our function is also going to approach zero now let's just look at a go let's just graph this or visualize this to get a sense of that so this is our x-axis whoops that's our x-axis let's call that ru ru access maybe and I did you intentionally because that's the variable we'll use in our proof of the chain rule video and let's say this right over here is our function let's say this right over here that is C this right over here is U of C U of C and then let's just take some arbitrary x over here so some arbitrary X and then this right over here this right over here is U of X is U of X so if we define if we define our change if we define let me do this if we define our change in our change in U is equal to U of X minus U of C which makes sense because this is our change in you so let's say this is going to be U of X minus U of C and if we define our change in X is equal to X minus C which is in this case it is X minus C it is X minus C then we can rewrite this limit right over here instead of saying the limit as X approaches C we could write the limit as Delta X approaches 0 because if X approaches C then Delta X is going to approach 0 so we could write this the limit as Delta X approaches 0 of Delta U of Delta U is going to be equal to 0 this we define this as our change in U and it is our change in U so this is equal to 0 so another way of thinking about this is as Delta X approaches 0 our change in U our change in the function is going to approach 0 so as as Delta X approaches 0 as Delta X approaches 0 Delta u approaches 0 and that's what we wrote over here Delta U approaches 0 as Delta X approaches 0 and a lot of ways this is hopefully a common sense we're dealing with a continuous function as you get smaller and smaller and you could just think of it this way as you get smaller and smaller changes in X's as our change in X gets smaller and smaller and smaller and smaller well because it's continuous you wouldn't be able to say this for a discontinuous function but because it's continuous or you wouldnt be able to say this for for some just can use discontinuous functions as our change in X gets smaller and smaller and smaller then our change in U is going to get smaller and smaller and smaller so it makes intuitive sense but hopefully this makes you feel even better about it because we're going to use this idea to prove the chain rule in the next video

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