Intermediate value theorem | Existence theorems | AP Calculus AB | Khan Academy

What we're going to cover in this video is the Intermediate Value Theorem, which despite some of this mathy language, you'll see is one of the more intuitive theorems, possibly the most intuitive theorem you will come across in a lot of your mathematical career. So first, I'll just read it out and then I'll interpret it, and hopefully, we'll all appreciate that it's pretty obvious. I'm not going to prove it here, but I think the conceptual underpinning here should be straightforward.

So the theorem tells us to suppose ( f ) is a function continuous at every point of the interval, the closed interval. So we're including ( A ) and ( B ), so it's continuous at every point of the interval ( [A, B] ). And so, let me just draw a couple of examples of what ( f ) could look like just based on these first lines. So suppose ( f ) is a function continuous at every point of the interval ( [A, B] ). So let me draw some axes here. So that's my ( y )-axis, and this is my ( x )-axis.

So one situation, if this is ( A ) and this is ( B ), ( f ) is continuous at every point of the closed interval ( [A, B] ). So that means it's got to be for sure defined at every point as well as being continuous. To be continuous, you have to be defined at every point, and the limit of the function as you approach that point should be equal to the value of the function at that point.

The function is definitely going to be defined at ( f(A) ), so it's definitely going to have an ( f(A) ) right over here. This right over here is ( f(A) ); maybe ( f(B) ) is higher, although we can look at different cases. So that would be our ( f(B) ), and they tell us it is a continuous function. It is a continuous function.

So if you're trying to imagine continuous functions, one way to think about it is if we're continuous over an interval, we take the value of the function at one point of the interval, and if it's continuous, we need to be able to get to the other, the value of the function at the other point of the interval without picking up our pencil.

So I can do all sorts of things; it still has to be a function. So I can't do something like that, but as long as I don't pick up my pencil, this is a continuous function. So there you go. If I had to—if I—if I—if the somehow the graph I had to pick up my pencil, if I had to do something like this, oops, I got to pick up my pencil and do something like that, well that's not continuous anymore. If I had to do something like this and “oops” pick up my pencil, not continuous anymore. If I had to do something like “whoop,” oh okay, pick up my pencil, go down here, not continuous anymore.

So this is what a continuous function that is a function that is continuous over the closed interval ( [A, B] ) looks like. I can draw some other examples; in fact, let me do that. So let me draw one maybe where ( f(B) ) is less than ( f(A) ). So it's my ( y )-axis, and this is my ( x )-axis, and once again, ( A ) and ( B ) don't both have to be positive; they could both be negative. One could be ( A ) could be negative, ( B ) could be positive, and maybe in this situation, ( f(A) ) and ( f(B) ) could also be positive or negative.

But let's take a situation where this is ( f(A) ); so that right over there is ( f(A) ), this right over here is ( f(B) ). And once again, we're saying ( f ) is a continuous function, so I should be able to go from ( f(A) ) to ( f(B) ) draw a function without having to pick up my pencil. So it could do something like this—actually, I don't want to make it go vertical; it could go like this and then go down and then do something like that.

So these are both cases, and I could draw an infinite number of cases where ( f ) is a function continuous at every point of the closed interval from ( A ) to ( B ). Now, given that there are two ways to state the conclusion for the Intermediate Value Theorem, you'll see it written in one of these ways or something close to one of these ways, and that's why I included both of these.

So one way to say it is, well, if this first statement is true, then ( f ) will take on every value between ( f(A) ) and ( f(B) ) over the interval. And you see in both of these cases, every interval—sorry, every value between ( f(A) ) and ( f(B) )—so every value here is being taken on at some point. You can pick some value; you could pick some value, an arbitrary value ( L ) right over here. Well, look, ( L ) happened right over there. If you pick ( L ), well ( L ) happened right over there, and it also happened there, and it also happened there.

And the second bullet point describes the Intermediate Value Theorem more in that way: for any ( L ) between the values of ( f(A) ) and ( f(B) ), there exists a number ( c ) in the closed interval from ( A ) to ( B ) for which ( f(c) = L ), so there exists at least one ( C ). So in this case, that would be our ( C ) over here. There's potential; there are multiple candidates for ( C ); that could be a candidate for ( C ); that could be a ( c ).

So we could say there exists at least one number—at least one number. I'll throw that in there—at least one number ( c ) in the interval for which this is true. And you know something that might amuse you for a few minutes is try to draw a function where this first statement is true, but somehow the second statement is not true. So you say, okay, well, let's say let's assume that there's an ( L ) where there isn't a ( c ) in the interval.

Well, let me try to do that. I'll draw, and I'll draw it big so that we can really see how obvious that we have to take on all of the values between ( f(A) ) and ( f(B) ) is. So let me draw a big axis this time. So that's my ( y )-axis, and that is my ( x )-axis, and I'll just do the case where, just for simplicity, that is ( A ) and that is ( B ), and let's say that this is ( f(A) ); so that is ( f(A) ), and let's say that this is ( f(B) ); so dotted line—alright, ( f(B) )—and we assume that we have a continuous function here.

So the graph I could draw it from ( f(A) ) to ( f(B) ), from this point to this point, without picking up my pencil from this coordinate ( (A, f(A)) ) to this coordinate ( (B, f(B)) ) without picking up my pencil. Well, let's assume that there is some ( L ) that we don't take on. Let's say there's some value ( L ) right over here, and we never take on this value. This continuous function never takes on this value as we go from ( x ) equaling ( A ) to ( x ) equal ( B ).

Let's see if I can draw that. Let's see if I can get from here to here without ever essentially crossing this dotted line. Well, let's see—I could, maybe I'll avoid it a little bit. I'll avoid, but gee, how am I going to get there without picking up my pencil? Well, well, I really need to cross that line. Alright, well, there you go! I found we took on the value ( L ), and it happened at ( C ), which is in that closed interval.

So once again, I'm not giving you a proof here, but hopefully, you have a good intuition that the Intermediate Value Theorem is kind of common sense. The key is you're dealing with a continuous function. If you were to make its graph, if you were to draw it between the coordinates ( (A, f(A)) ) and ( (B, f(B)) ), and you don't pick up your pencil—which would be true of a continuous function—you're going to take on every value between ( f(A) ) and ( f(B) ).