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# Intermediate Value Limit Theorem Proof, Example

The intermediate value theorem illustrates that for each value connecting the least upper bound and greatest lower bound of a continuous curve, where one point lies below the line and the other point above the line, and there will be at least one place where the curve crosses the line.

## Intermediate Value Limit Theorem Proof, Example

 Statement: If f(x) is a continuous function on [a, b] then for every d between f (a) and f (b), there exists c between a and b so that f(c) =d. Proof: Suppose f is a continuous function on [a, b]. Let v be a real number between f (a) and f (b). Since f is continuous, it takes on every number between f (a) and f (b), i.e., every intermediate value. Thus, d is a value of f. From the above diagram, If d is between f(a) and f(b) then there exists c ,between a and b, so that f(c)=d. Let's partition [a, b] into two sets of A and B such that, each number in [a, b] is in either A or B and each number in A is less than every number in B. Suppose if f(a)< f(b) then a parallel argument will lead to the same conclusion if f (a)>f(b).Let α be in [a, b] such that for all x in [a, α],f(x) <= d. Let A be the set of all such α's and B be the set of points in [a, b] which are not in A, then there exists x in [a, β] such that f(x)>d. So α< x <=β. Therefore each element of A is less than every element of B. Assume f(c) < d, By continuity, we have limx→d f(x) = f(c).Hence there exists δ> 0 such that 0 < |x-c| < δ => |f(x)-f(c)| < d-f(c) => f(x)-f(c) < d-f(c) => f(x)< d It follows that, since (f(x) ≤ d if x is in A if x < c), f(c) < d, and (0 < |x-c| < δ if c < x ≤ c + δ/2), we obtain like this: x ≤ c + δ/2 => f(x) < d. Therefore, (c+δ/2) is in A, which means that c is in A but is not the largest number in A, which is a contradiction. So we must have f(c) ≥ d . For example, assume f(c) > d, there exists δ > 0 such that: 0 < |x-c| < δ => |f(x)-f(c)| < f(c)-d => f(c)-f(x) < f(c)-d => f(x)>d In particular: c - δ < x ≤ c - δ/2 => f(x) > d. Because there is x ≤ c - δ/2 such that f(x)> d, (c-δ/2) is not in A, it is in B which means that c is in B but is not the smallest number in B, which is contradiction. As a consequence, we must have f(c)≤d. Hence we must have f(c) = d. Hence the Intermediate Value Limit Theorem is proved.

The intermediate value theorem illustrates that for each value connecting the least upper bound and greatest lower bound of a continuous curve, where one point lies below the line and the other point above the line, and there will be at least one place where the curve crosses the line.