Lecture 14: Operations and properties of continuity

14.1 Basic Concepts

14.1.1 Calculation Theorem

#####Theorem: Thecomputationalpropertiesofcontinuity

description:

1. The sum, difference, product, and quotient of continuous functions (with denominators not zero) are still continuous functions;

2. Thecompositeof continuous functions is still a continuous function;

3.Basic elementary functionsare continuous within theirdomain;

4.Elementary functionsare continuous within theirdefinition intervals;

Explanation

• Note:
• The fundamental elementary function is continuous within itsdomain;
• The elementary function is continuous within itsdefinition interval;
• Defined interval: Everything contained within the domain is defined interval;
• domain is unique, and the defined interval is not unique;
• Example: $1/x$, all values except for x=0 are domain; $(0,1)$ and $(1,+∞)$ are two independent defined intervals;