Lecture 15: Properties of continuous functions on closed intervals
Postgraduate Entrance Exam Mathematics Study Notes: Lecture 15: Properties of Continuous Functions on Closed Intervals. Retain original formulas, diagrams, and example problems.
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15.1 Basic Concepts
15.1.1 Boundedness
#####Theorem: Bounded
description: $\text{ If f(x) is continuous on [a,b], then f(x) is bounded on [a,b]. }$
Explanation
- Continuous values on a finite closed interval must have a maximum and a minimum value;
15.1.2 Extremes and Intermediate Values
#####Theorem: TheMostValuableTheorem
Description: if $f(x)$ is continuous on $[a,b]$, then $f(x)$ must have a maximum and minimum value on $[a,b]$;
#####Theorem: Theintervalencetheorem
Description: If $f(x)$ is continuity on $[a, b]$, and $f(a)\neq f(b)$, then there is at least one $\xi\in(a, b), $ for any number of $\mathbf{C}, $ between $f(a)$ and $f(b)$ such that $f(\xi){=}C.$
15.1.3 Zero point
#####Theorem: Zeropointtheorem
description: if $f(x)$ is continuous on $[a,b]$ and $f(a)\cdot f(b)<0$, then $\exists\xi\in(a,b)$ must be such that $f(\xi)=0.$