Chapter 1: Limits
Section 1.7: Intermediate Value Theorem
In essence, the Intermediate Value theorem states that a continuous function f takes on every value between fa and fb. This property of a continuous function is called the "Darboux" property. Hence, every continuous function has the Darboux property, but there are functions that have the Darboux property that are not continuous. As a counterexample, take the function whose rule is sin1/x for x≠0, and has the value 0 at x=0. This function takes on every value between −1 and 1, yet is not continuous.
As intuitive as the Intermediate Value theorem (IVT) might seem, it is equivalent to the far deeper completeness property of the real numbers, which essentially states that there are no gaps in the real number line. If there were, then the IVT might not hold because a crucial number could be missing from the range of f. So, asserting that f takes on every possible value between fa and fb is an assertion that there are no gaps in the real line between these two values, that is, the real numbers are "complete."
For more than a century after Newton and Leibniz articulated the tools of the calculus, these tools were applied to a host of problems in physics and engineering, with spectacular success. It wasn't until much later that mathematicians began to consider such foundational issues as the IVT and its relation to completeness of the real numbers. However, theorems like the IVT are behind interesting observations even today. Click here for an article discussing the use of the IVT in proving that a "wobbly table" can be stabilized by a rotation about its center.
The Intermediate Value Theorem
A formal statement of the IVT is the following.
Theorem 1.7.1: The Intermediate Value Theorem
Let f be a continuous function defined on a closed interval [a,b] and let z be a number between fa and fb . The equation fc=z has at least one solution c in the open interval ⁡(a,b).
For an alternate statement of the Intermediate Value theorem, consider
A continuous function on a closed interval must attain every value between the function values at the endpoints.
Stated this way, the IVT might sound "obvious," but either way, it is actually a deep theorem because it is equivalent to the completeness of the real numbers, itself a deep characterization of the reals.
The IVT is the theoretical basis for the Bisection method, a linearly converging numeric technique for solving algebraic equations. This computational method is detailed in Example 1.7.1.
The IVT is also the key in Examples 1.7.2 and 1.7.3.
Use bisection to approximate the zeros of f⁡x=x5−4⁢x2+2x−1.
Suppose the temperature at midnight is 55°F, increases to a high of 85°F, then returns to 55°F at midnight the next day. Assuming that the temperature throughout the day is a continuous function of the time of day, show that there is at least one time in the morning when the temperature is the same as the temperature exactly 12 hours later.
A simple closed path is walked in rugged terrain. Suppose the path is parametrized by s, with 0≤s≤1. and suppose further that the heights along the path are given by hs. Prove that there are two points along the path where ha+1/2=ha.
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