![]() ![]() You can pick some value,Īn arbitrary value L, right over here. So every value here is being taken on at some point. And you see in both of these cases every interval, sorry, every every value between F of A and F of B. So one way to say it is, well if this first statement is true then F will take on every value between F of A and F of B over the interval. You'll see it written in one of these ways or something close to one of these ways. Now, given that there's two ways to state the conclusion for the intermediate value theorem. Infinite number of cases where F is a functionĬontinuous at every point of the interval. So these are both cases and I could draw an So I should be able to go from F of A to F of B F of B draw a function without having to pick up my pencil. And once again we're saying F is a continuous function. But let's take a situation where this is F of A. And F of A and F of B it could also be a positive or negative. And once again, A and B don't both have to be positive, they can both be negative. I can draw some other examples, in fact, let me do that. So, this is what a continuous function that a function that is continuous over the closed interval A, B looks like. Pencil, go down here, not continuous anymore. If I had to do something like this and oops, pick up my pencil not continuous anymore. Pencil do something like that, well that's not continuous anymore. If I had to do something like this oops, I got to pick up my If the somehow the graph I had to pick up my pencil. But, as long as I don't pick up my pencil this is a continuous function. So, I can do all sorts of things and it still has to be a function. Value of the function at the other point of the interval without picking up our pencil. ![]() We need to be able to get to the other, the 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 they tell us it isĪ continuous function. So it's definitely going to have an F of A right over here. And so the function isĭefinitely going to be defined at F of A. That is recorded at that point should be equal to the value As well, as to be continuous you have to defined at every point. Point of the interval of the closed interval A and B. Suppose F is a functionĬontinuous at every point of the interval A, B. Let me just draw a couple of examples of what F could look like just based on these first lines. That suppose F is a function continuous at every point of the interval the closed interval, so Underpinning here is it should be straightforward. So first I'll just read it out and then I'll interpret it and hopefully we'll all appreciate Intuitive theorem you will come across in a lot of your mathematical career. Mathy language you'll see is one of the more intuitive theorems possibly the most Gonna cover in this video is the intermediate value theorem. ![]()
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