5.3 Continuous Functions on Intervals

<< 5.2 Combinations of Continuous Functions

5.3.1 Definition

A function $f:A\to \mathbb{R}$ is said to be bounded on $A$ if there exists a constant $M>0$ such that $|f(x)|\le M$ for all $x\in A$.

5.3.2 Boundedness Theorem

Let $I:=[a,b]$ be a closed bounded interval and let $f:I\to \mathbb{R}$ be continuous on $I$. Then $f$ is bounded on $I$.

5.3.3 Definition

Let $A\subseteq \mathbb{R}$ and let $f:A\to \mathbb{R}$. We say that $f$ has an absolute maximum on $A$ if there is a point $x^{\ast }\in A$ such that $f(x^{\ast })\ge f(x)$ for all $x\in A$. We say that $f$ has an absolute minimum on $A$ if there is a point $x_{\ast }\in A$ such that $f(x_{\ast })\le f(x)$ for all $x\in A$. We say that x^ is an absolute maximum point for $f$ on $A$, and that x_ is an absolute minimum point for $f$ on $A$, if they exist.

5.3.4 Maximum-Minimum Theorem

Let $I:=[a,b]$ be a closed bounded interval and let $f:I\to \mathbb{R}$ be continuous on $I$. Then $f$ has an absolute maximum and an absolute minimum on $I$.

5.3.5 Location of Roots Theorem

Let $I=[a,b]$ and let $f:I\to \mathbb{R}$ be continuous on $I$. If $f(a)<0<f(b)$, or if $f(a)>0>f(b)$, then there exists a number $c\in (a,b)$ such that $f(c)=0$.

5.3.7 Bolzano’s Intermediate Value Theorem

Let $I$ be an interval and let $f:I\to \mathbb{R}$ be continuous on $I$. If $a,b\in I$ and if $k\in \mathbb{R}$ satisfies $f(a)<k<f(b)$, then there exists a point $c\in I$ between $a$ and $b$ such that $f(c)=k$.

5.3.8 Corollary

Let $I=[a,b]$ be a closed, bounded interval and let $f:I\to \mathbb{R}$ be continuous on $I$. If $k\in \mathbb{R}$ is any number satisfying $\inf f(I)\le k\le \sup f(I)$, then there exists a number $c\in I$ such that $f(c)=k$.

5.3.9 Theorem

Let $I$ be a closed bounded interval and let $f:I\to \mathbb{R}$ be continuous on $I$. Then the set $f(I):=f(x):x\in I$ is a closed bounded interval.