Properties of Continuous Functions

Primary tabs

Lesson Summary: 
We summarize the properties of continuous functions.
Lesson Inputs: 
Bounded and Unbounded Functions
Function Limits V: Properties of Function Limits
Function Limits VI: Continuous Functions
Geometry of Functions II: The Extreme-Value Theorem
The Intermediate-Value Theorem

SAMPLE LESSON

Sign in for easy to read lessons narrowly focused on a specific subject. All lessons are linked to each other allowing you to easily see the connections between different Calculus concepts. Our unique Calcmap will give you a bird's-eye view of all our lessons and help you see how important concepts are related to each other. All of the formulas are available to our subscribers, but here is a sample of the lesson you will see when you log in:



Lesson: 

Here are some important aspects of continuous functions:

  1. Continuous functions are bounded.
  2. The Intermediate-Value Theorem proves our original intuition that a continuous function is one that can be drawn on a piece of paper without having to lift the pencil from the paper to draw it.
  3. The Extreme-Value Theorem describes how a function will take on both a maximum value and a minimum value on [a,b].
  4. The composition of two continuous functions is a continuous function.

We can also use our properties of function limits to show that when f(x) and g(x) are continuous,

<Sign in to see all the formulas>


then the follow properties are true:

Properties of Continuous Functions Obtained From the Properties of Function Limits
A constant times a continuous function is a continuous function:

<Sign in to see all the formulas>

A constant plus a continuous function is a continuous function:

<Sign in to see all the formulas>

The sum of two continuous functions is a continuous function:

<Sign in to see all the formulas>

The product of two continuous functions is a continuous function:

<Sign in to see all the formulas>

The quotient of two continuous functions is a continuous function:

<Sign in to see all the formulas>

Polynomials are continuous:

<Sign in to see all the formulas>