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# Intermediate Value Theorem

## More Formal

❶Then is nonempty since and it has an upper bound namely so there is a least upper bound.   For this function there is an for any value between and. For two real numbers and with , let be a continuous function on the closed interval Then for every between and there exists a number with. A continuous function attaining the values and also attains all values in between. The idea of the proof is as follows: These points demarcate a segment of the graph which connects the two points, as in the picture above. Since this segment has no holes or breaks in it by continuity of the function , it must pass through each "level curve" where.

Note that the guaranteed by the theorem may not be unique; the theorem implies that there is at least one such that but there may be more than one, depending on and. The statement of the theorem has multiple requirements, all of which are necessary for the conclusion to hold.

Here is an illustrative example:. For the function , over which of the following intervals does the intermediate value theorem guarantee a root:. In order for the intermediate value theorem to guarantee a root on a specified interval , not only must the function be continuous on the interval, but 0 must be contained between and Let's check the values of and. For the first interval the values returned by are both positive which do not sandwich 0, meaning the intermediate value theorem does not guarantee a root.

For the second interval the values returned by are on either side of 0, which seems to suggest that has a root on the interval However, it's important to note that has a discontinuity at meaning the intermediate value theorem does not hold. Indeed, does not have an -intercept on the interval. For the last interval the values returned by are on either side of 0, which implies that has a root on the interval This is confirmed by the intermediate value theorem because is continuous on.

Graph of f x. As in the above example, one simple and important use of the intermediate value theorem hereafter referred to as IVT is to prove that certain equations have solutions.

Consider the following example:. Does the equation have solution s? If so, how many solutions does it have? We study the function.

Thus, by the IVT, there must be some such that , i. One root of the equation has been identified. Is this the only root?

Note that is everywhere non-negative, so is increasing monotonically. Hence, can only have one root. Is there a solution to where. At , we have At , we have. So the IVT implies that there is a solution to in the interval. Suppose that is continuous on and. Let be any positive integer, then prove that there is some number such that. Consider the set of numbers. Let be such that is the largest number in.

By the Intermediate value theorem, there is with , so that , or as desired. Finally, if the largest number in is , then the same argument works with chosen such that is the minimum number in. Note that if is both the largest and smallest number in , then they are all the same and. Let be the hyperreal unit, then prove that there is some number such that. First, assume is not constant on.

The result holds trivially if it is. Since is not constant, there exists a such that is a maximum or a minimum, but assume for now that it is a max; a min is handled similarly. Now, by the Intermediate Value Theorem, if , there exists a such that. Thus, , as wanted. The equation has a unique real solution. Yes No A real-valued function is said to have the intermediate value property if for every in the domain of , and for every.

The intermediate value theorem states that if is continuous, then has the intermediate value property. Is the converse of this theorem true? That is, if a function has the intermediate value property, must it be continuous on its domain? Since it can detect zeroes of functions, the IVT is an important tool for the analysis of continuous functions.

However, through some clever contortions, IVT can give even more impressive results. For instance, one can prove the Borsuk-Ulam theorem in dimension 1. This theorem states that for any continuous real-valued function on a circle, there is some point on the circle such that takes the same value at and at the point on the circle directly opposite to the antipode of.

This implies that on any great circle of the globe, any continuously varying information will take on the same value at some two antipodal points. For instance, there must exist two antipodal points on the equator at which the air temperature is the same. This kind of discontinuity in a graph is called a jump discontinuity.

Jump discontinuities occur where the graph has a break in it as this graph does and the values of the function to either side of the break are finite i. The function is not continuous at this point.

This kind of discontinuity is called a removable discontinuity. Removable discontinuities are those where there is a hole in the graph as there is in this case.

A function is continuous on an interval if we can draw the graph from start to finish without ever once picking up our pencil. The graph in the last example has only two discontinuities since there are only two places where we would have to pick up our pencil in sketching it. Rational functions are continuous everywhere except where we have division by zero. So all that we need to is determine where the denominator is zero.

With this fact we can now do limits like the following example. Below is a graph of a continuous function that illustrates the Intermediate Value Theorem. Also, as the figure shows the function may take on the value at more than one place. It only says that it exists. These are important ideas to remember about the Intermediate Value Theorem.

A nice use of the Intermediate Value Theorem is to prove the existence of roots of equations as the following example shows. ## Main Topics

Intermediate value theorem. (Version I). Consider a closed interval = [,] in the real numbers and a continuous function: →. Then, if is a real number such that .

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The idea behind the Intermediate Value Theorem is this: When we have two points connected by a continuous curve: one point below the line. the other point above the line.