The Intermediate Value Theorem states that if a function is continuous on the closed interval

Note: The Intermediate Value Theorem is an existence theorem. It tells you that a number exists, but does not explicitly tell you the number.

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*(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