Continuity in Calculus: Definition, Examples & Problems

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  • 0:03 Continuous Function
  • 0:39 Limits Review
  • 1:31 What Is Continuity?
  • 3:15 Examples Using Graphs
  • 5:18 Examples Using Equations
  • 6:11 Lesson Summary
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Lesson Transcript
Instructor: Megan Robertson
Graphing functions can be tedious and, for some functions, impossible. Calculus gives us a way to test for continuity using limits instead. Learn about continuity in calculus and see examples of testing for continuity in both graphs and equations.

Continuous Function

At the basic level, teachers tend to describe continuous functions as those whose graphs can be traced without lifting your pencil. While it is generally true that continuous functions have such graphs, this is not a very precise or practical way to define continuity. Many graphs and functions are continuous, or connected, in some places, and discontinuous, or broken, in other places. There are even functions containing too many variables to be graphed by hand. Therefore, it's necessary to have a more precise definition of continuity, one that doesn't rely on our ability to graph and trace a function.

Limits - A Quick Reminder

The definition of continuity in calculus relies heavily on the concept of limits. In case you are a little fuzzy on limits: The limit of a function refers to the value of f(x) that the function approaches near a certain value of x.

The limit of a function as x approaches a real number a from the left is written like this:

Left limit notation

The limit of a function as x approaches a real number a from the right is written like this:

Right Limit Notation

If the left limit and the right limit exist (are not infinity) and are equal, then we say the limit of the function as x approaches a exists and is equal to the one-sided limits. We write it like this:

General Limit Notation

Remember, the limit describes what the function does very close to a certain value of x. The function value at the point x = a is written f(a).

What Is Continuity?

In calculus, a function is continuous at x = a if - and only if - all three of the following conditions are met:

  1. The function is defined at x = a; that is, f(a) equals a real number
  2. The limit of the function as x approaches a exists
  3. The limit of the function as x approaches a is equal to the function value at x = a

There are three basic types of discontinuities:

  1. Removable (point) discontinuity - the graph has a hole at a single x-value. Imagine you're walking down the road, and someone has removed a manhole cover (Careful! Don't fall in!). This function will satisfy condition #2 (limit exists) but fail condition #3 (limit does not equal function value).
  2. Infinite discontinuity - the function goes toward positive or negative infinity. Imagine a road getting closer and closer to a river with no bridge to the other side
  3. Jump discontinuity - the graph jumps from one place to another. Imagine a superhero going for a walk: he reaches a dead end and, because he can, flies to another road.

Both infinite and jump discontinuities fail condition #2 (limit does not exist), but how they fail is different. Recall for a limit to exist, the left and right limits must exist (be finite) and be equal. Infinite discontinuities have infinite left and right limits. Jump discontinuities have finite left and right limits that are not equal.

Examples Using Graphs

Let's go through some examples using this graph to represent the function of f(x):

Discontinuous Graph y=f(x)

Example 1

Is f(x) continuous at x = 0?

Graph of y=f(x) near x=0

To check for continuity at x = 0, we check the three conditions:

  1. Is the function defined at x = 0? Yes, f(0) = 2
  2. Does the limit of the function as x approaches 0 exist? Yes
  3. Does the limit of the function as x approaches 0 equal the function value at x = 0? Yes

Since all three conditions are met, f(x) is continuous at x = 0.

Example 2

Is f(x) continuous at x = -4?

Graph of y = f(x) near x = -4

To check for continuity at x = -4, we check the same three conditions:

  1. The function is defined; f(-4) = 2
  2. The limit exists
  3. The function value does not equal the limit; point discontinuity at x = 4

Example 3

Is f(x) continuous at x = -2?

Graph of y=f(x) near x=-2

Here are those conditions again:

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