Continuous Functions

1. Continuous Functions
1. Intervals
   For two real numbers and ( ), the sets
   
   
   
   
   are called intervals, and they are represented as
   
   
   
   , respectively.
   Here, is called a closed interval, is called an open interval, and and are called half-closed intervals or half-open intervals.
2. Continuous Function
   A function is considered continuous on an interval if it is continuous for all real numbers within that interval. Specifically, is continuous on the closed interval if it satisfies the following conditions:
1. It is continuous on the open interval .
2. ,


2. Properties of Continuous Functions
   If two functions and are continuous at , then the following functions are also continuous at :
1. (where is a constant)
4. (where )