Intermediate Value Theorem

If is continuous on a closed interval [a, b] , and is any number between f(a)andf(b) inclusive, then there is at least one number in the closed interval such that f(x)=c .

The theorem is proven by observing that f([a,b]) is connected because the image of a connected set under a continuous function is connected, where f([a,b]) denotes the image of the interval under the function . Since c is between f(a) and f(b) , it must be in this connected set.

A function that is continuous on an interval has no gaps and hence cannot "skip over" values. If a function is continuous on a closed interval from x = a to x = b, then it has an output value for each x between a and b. In fact, it takes on all the output values between f (a) and f (b); it cannot skip any of them. More formally, the Intermediate Value Theorem says:

Let f be a continuous function on a closed interval ([a,b]) . If k is a number between f (a) and f (b), then there exists at least one number c in [a,b] such that f (c) = k.

In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is a corresponding point in its domain that the function maps to that value. The theorem depends on (and is actually equivalent to) the completeness of the real numbers. It is false for the rational numbers Q.

Simply put, the theorem states that if we have a continuous function and two points on the function, then we can connect the two points with a line. If I give you a y-value between the y-value of the two points, then you can tell me the corresponding x-value for it.