- Topics At a Glance
- Continuity at a Point
- Continuity at a Point via Pictures
- Continuity at a Point via Formulas
- Functions and Combinations of Functions
- Continuity on an Interval
- Continuity on an Interval via Pictures
- Continuity on an Interval via Formulas
- Continuity on Closed and Half-Closed Intervals
- Determining Continuity
- The Informal Version
- The Formal Version
- Properties of Continuous Functions
- Boundedness
- Extreme Value Theorem
- Intermediate Value Theorem
- In the Real World
- Page: I Like Abstract Stuff; Why Should I Care?
- Page: How to Solve a Math Problem
- Appendix: Intervals and Interval Notation
Introduction to :
When we say a function f is continuous, we usually mean it's continuous at every real number. In other words, it's continuous on the interval (-∞, ∞).
Some examples of continuous functions that are continuous at every real number are: polynomials, ex, sin(x), and cos(x).
If we add, subtract, multiply, or compose continuous functions we find new continuous functions. If we take a quotient of continuous functions 
, this quotient will be continuous on any intervals that do not include places where g is zero.
Practice:
Example 1
Let f(x) = 4x2 + 3x and g(x) = sin(x). Determine whether each function is continuous. If not, where is the function discontinuous?
|
is continuous wherever g is not zero. g is zero at all multiples of π (0,π, -π, 2π,-2π, ...). 
will be continuous on the intervals ..., (-2π, π), (-π,0), (0,π), (π,2π),...
is continuous wherever f is not zero. 
, the function 
is continuous on the intervals 
