Function Limits VI: Continuous Functions

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Lesson Summary: 
We define both an intuition and a rigorous mathematical definition of continuous functions.
Lesson Inputs: 
Function Limits V: Properties of Function Limits
Function Limits III: The Gory Details
Lesson Outputs: 
Antiderivatives
Geometry of Functions II: The Extreme-Value Theorem
Differentiation III: How Differentiability and Continuity Are Related
A List of Continuous Functions
The Intermediate-Value Theorem
Properties of Continuous Functions
Function Limits VII: Putting It All Together With Useful Examples

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Lesson: 

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. For example,

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is an example of a continuous function.


An example of a continuous function.

However,

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is a function that cannot be drawn on a piece of paper without lifting your pencil. At x=1 the function has a value of 1 and immediately to the right of x=1 the function has a value of 5! This is definitely not a continuous function because we must lift our pencil off of the sheet at x=1 to continue drawing the function. Note that this function is continuous at x=2.


An example of a discontinuous function.

Furthermore, the functions

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are both discontinuous at x=1 because they are not defined there at all. Note that the second function is identical to our first example except at x=1.

The official mathematical definition of continuity is that a function is continuous at a point x=a when

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We have always said that a function limit

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does not have to equal the function value at x=a. In other words, L does not have to equal f(a). But for a continuous function, <Sign in to see all the formulas>. This is how we can draw a picture of a function and not have to lift up our pencil.

Similar to one-sided limits, we can define one-sided continuity:

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and

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But we know that <Sign in to see all the formulas> if and only if <Sign in to see all the formulas> and <Sign in to see all the formulas>. Therefore, for continuous functions,

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The left-sided and right-sided function limits must equal f(a) for a continuous function.

Finally, here are some important properties of continuous functions.