Intermediate Value Theorem
This theorem tells us that continuous functions don't jump. They have to attain every value between two values in their image.
Discover Bridges
Study Intermediate Value Theorem
Theorem 1. Intermediate Value Theorem Let \(a,b\in\mathbb{R}\) with \(a<b\). Moreover, let \(f:[a,b]\to\mathbb{R}\) be continuous. Let \(x_0,x_1\in [a,b]\) and let \(y\in\mathbb{R}\) be between \(f(x_0)\) and \(f(x_1)\). Then there exists some \(\hat{x}\) between \(x_0\) and \(x_1\) such that \(y=f(\hat{x})\).
Proof. Without loss of generality, we assume that \(x_0<x_1\). Moreover, we can assume
without loss of generality that \(y=0\)
(otherwise, consider the function \(f(x)-y\) instead of \(f\)). Furthermore, we can assume without
loss of generality that \(f(x_0)\leq0\)
and \(f(x_1)\geq0\) (otherwise,
consider \(-f\) instead of \(f\)).
We will construct \(\hat{x}\) by nested
intervals (compare the proof of the Bolzano-Weierstrass Theorem).
Inductively define \(A_0=x_0\), \(B_0=x_1\) and for \(k\geq1\),
\(A_k=A_{k-1}\), \(B_k=\frac{A_{k-1}+B_{k-1}}2\), if \(f(\frac{A_{k-1}+B_{k-1}}2)\geq0\), and
\(A_k=\frac{A_{k-1}+B_{k-1}}2\), \(B_k=B_{k-1}\), if \(f(\frac{A_{k-1}+B_{k-1}}2)\leq0\).
Then we have \[\lim_{n\to\infty}A_n=\lim_{n\to\infty}B_n=:\hat{x}.\] By the continuity of \(f\) and the fact that \(f(A_n)\leq0\), \(f(B_n)\geq0\) for all \(n\in\mathbb{N}\), we have \[0\geq\lim_{n\to\infty}f(A_n)=f(\hat{x})=\lim_{n\to\infty}f(B_n)\geq0\] and thus \(f(\hat{x})=0\). ◻
An episode by Fabian Gabel and Julian Großmann.
Discuss your questions by typing below.
Solve the WeBWorK Exercise
The data for the interactive network on this webpage was generated with pntfx Copyright Fabian Gabel and Julian Großmann. pntfx is licensed under the MIT license. Visualization of the network uses the open-source graph theory library Cytoscape.js licensed under the MIT license.