The Intermediate Value Theorem states that if a function is continuous on the closed interval (a,b) , and k is any number between f(a) and f(b), then there is at least one number c in the closed interval such that f(c)=k. 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 (a,b) under the function . Since k is between f(a) and f(b), it must be in this connected set. Proof of this theorem is based on a property of real numbers called completeness. The theorem states that for a continuous function f, if x takes on all values between a and b, f(x) must take on all values between f(a) and f(b).
Note: The Intermediate Value Theorem is an existence theorem. It tells you that a number exists, but does not explicitly tell you the number.
Source: Calculus of a Single Variable Sixth Edition book
Note: The Intermediate Value Theorem is an existence theorem. It tells you that a number exists, but does not explicitly tell you the number.
Source: Calculus of a Single Variable Sixth Edition book