In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and largest 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 upon the completeness of real numbers. It is not true for rational numbers Q.

Put the theorem states that if we have a continuous function and 2 points on the function, then we can connect the 2 points with a line. If I give you a y-value between the y-value of the 2 points, then you can tell me the corresponding x-value for it.

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

Let f be the continuous function on closed interval [a,b]. If k is a number in between f (a) and f (b), then there exists at least 1 number c in [a,b] such that f (c) = k.

If f is continuous on closed interval [a,b], and c is any number in between f (a), and f (b), inclusive, then there is at least 1 number x in the closed interval so that f (x)=c

The theorem is proved by observing that f ([a,b]), is connected as the image of a connected set under the continuous function is connected, here f ([a,b]), denotes the image of interval [a,b] under function f . As c is between f (a), and f (b),, it should be in this connected set.