Properties of Continuous Functions

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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

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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,

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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:

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A constant plus a continuous function is a continuous function:

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The sum of two continuous functions is a continuous function:

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The product of two continuous functions is a continuous function:

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The quotient of two continuous functions is a continuous function:

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Polynomials are continuous:

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