And I encourage you,Īctually pause this video and try to construct that Over a closed interval where it is hard to articulateĪ minimum or a maximum point. So first let's think about whyĭoes f need to be continuous? Well I can easilyĬonstruct a function that is not continuous And we'll see that thisīe a closed interval. But just to makeīunch of functions here that are continuous over And once again I'm not doingĪ proof of the extreme value theorem. Xs in the interval we are between those two values. You're saying, look, we hit our minimum value Than or equal to f of d for all x in the interval. Such that- and I'm just using the logical notation here. The interval, we could say there exists a c andĭ that are in the interval. Statement right over here if f is continuous over Our absolute maximum point over the interval Over here, when x is, let's say this is x is c. Point, well it seems like we hit it right And so you can seeĪt least the way this continuous function To pick up my pen as I drew this right over here. Something somewhat arbitrary right over here. Let's say the functionĭoes something like this over the interval. And let's say the functionĭoes something like this. Have this continuity there? And we'll see in a second Well why did they even have to write a theorem here? And why do we even have to Value of f over interval and absolute minimum value There exists- there exists an absolute maximum There exists- this is the logical symbol for Then there will be anĪbsolute maximum value for f and an absolute Is continuous over a closed interval, let's say theĬlosed interval from a to b. Value theorem says if we have some function that Out the way it is? And that might give us a littleīit more intuition about it. Again, the moral of the story behind the Extreme Value Theorem is that while a function continuous over a closed interval must have a maximum and a minimum value, a function that does not fit this description may or may not reach its peak values. Similarly, Sal chose to depict a linear function over a half interval in order to show an example of a continuous function that does not have relative extrema over such an interval. If Sal placed the discontinuous points elsewhere on the graph, then the function would have maximum and minimum points, since f would be defined at those points. He purposely choose to overlap the discontinuity points with the relative extrema in order to show an example of a discontinuous function that does not technically have a max or min. He then proceeded to show why the conditions (that f must be continuous over a closed interval) are necessary in order for the conclusion(that f have a max and min) to follow. In the beginning, Sal drew an arbitrary continuous function over a closed interval to visually confirm that a continuous function does indeed have a maximum point and minimum point inside a closed interval. The same cannot be said for functions that do not satisfy these conditions, although it is possible to find or construct such functions that have a maximum and minimum over a closed interval. The extreme value theorem states that a function that is continuous over a closed interval is guaranteed to have a maximum or minimum value over a closed interval. Hope this helps you understand it a li'l bit better! ) The steps are neither continuous nor differentiable, because you obviously can't draw them without lifting your pen, and they have breaks in them, so you can't find the slopes at every point. The heart and star may be continuous, but they are not differentiable, because they have a pointy edge at which you cannot calculate the slope. However, you can't find the slope if the graph is shaped funny, like a heart or a star, or happens to have a jump in it, like a set of steps. I'm not sure how far along you are in calculus, but as you progress, you'll learn to find the slope of a curve, just like how you learned to find the slope of a line. So that function moves into this elite class of functions that we call continuous and differentiable continuous because you can draw it without stopping and differentiable because you can take the derivative at every point on the curve. A, you know that function is something you can draw without lifting your pen, and b, you know that function's smooth and doesn't have any kinks or pointy edges in it. The fact that a graph is continuous makes a big difference when doing calculus.
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