If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Proof: Differentiability implies continuity

## Video transcript

what I hope to do in this video is prove that if a function is differentiable at some point see 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 differentia differentiability so let's think about that first it's always helpful to draw ourselves a function so that's our y-axis this is our 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 equal 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 is 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 what's going to 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 a review this is just a 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 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 con to new ax t 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 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 then 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 let me 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 so this this definition of continuity seems to be good at least for this case because this is not a discount this is this is not a continuous function you have a point discontinuity so for at least in this case are this this definition of continuity would properly identify this as not a continuous function now you could also think about a jump discontinuity you could also think about a jump discontinuity so let's look at this and all this is hopefully a little bit of review so jump discontinuity at C at x equals C might look like this might look 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 try to find evaluate the limit as X approaches C of f of X you get a different value as 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 a jump of this type of a jump discontinuity so once again this definition would properly say that this is not this 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 kind of do this review just so that you can really visualize things so differentiability implies this this this limit right over here exists so let's start with a slightly different limit let's limit let me draw a line here actually let me draw a line too 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 can 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 time divided 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 property this I'm gonna brought up by a property of limits here so the limit of the product is equal to the same thing as a product of the limits so the limit as X approaches C of X minus C times the limit let me write it 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 what is this thing right over here well if we assume that F is differentiable at C at C and we're going to do that actually I should have started off there let's assume let's assume because we wanted to start your show the differentiability improves continuity if we assume F differentiable differentiable at C well then this right over here is just the 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 going to be zero as X approaches C that's going to become C minus C is 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 do 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 parentheses around it like that right 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 I'll do it over here but actually let me 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 0 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 evaluate 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 0 we'll just add F of C to both sides and what you get will 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 the limit of my function is X approaches C is equal to the function is equal to the value of the function at C 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