In this unit, we explore what it means for a function to be continuous at a point and over an interval. A function f is considered to be continuous at x = c if all the following conditions are satisfied:
1. The limit of f(x) exists as x approaches x=c
2. f(c) is defined where x=c is in the domain of f(x)
3. f(c) = The limit of f(x) as x approaches x=c
On the other hand,
If f(x) is continuous at each point on an interval then it is continuous on an open interval. We say f is continuous on the closed interval [a, b] if it is continuous on (a, b), and right continuous at a, and left continuous at b. In other words, a function is continuous on an interval if it is continuous at every number in the interval.
The concept of function continuity and limits is essential in calculus.