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Continuity of Functions

Continuity of Functions

Page: How to Solve a Math Problem

There are three steps to solving a math problem.

  • Figure out what the problem is asking.
  • Solve the problem.
  • Check the answer.

Example. Determine all values of x at which the function 

is discontinuous.


  • Figure out what the problem is asking.

We want to make sure we understand the problem. What does discontinuous mean? It means "not continuous", but what does that mean?

A function is continuous at a value x = c if three things happen:

  • f(c) exists,
  •  exists, and

For the function to be discontinuous at x = c, one of the three things above need to go wrong. Either

  • f(c) is undefined,
  • doesn't exist, or
  • f(c) and  both exist, but they disagree.

This problem is asking us to examine the function f and find any places where one (or more) of the things we need for continuity go wrong.

  • Solve the problem.
  • Where is f(x) undefined?

Since we're looking at a rational function, f is undefined wherever its denominator is 0. To find where that is, we need to factor the numerator and denominator.

When the denominator is 0, either x = -3 or x = 4.

  • Where does  not exist?

This rational function has a hole at x = -3 and a vertical asymptote at x = 4, therefore doesn't exist. This gives us another reason that f(x) is discontinuous at x = 4.

  • Where do f(c) and  both exist, but disagree?

This function doesn't have any places like that! Since a rational function is continuous everywhere it's defined, we've found all the discontinuous places we need to worry about.

To summarize, this function is only discontinuous at x = -3 and x = 4.

  • Check the answer.

Besides doing the arithmetic again, probably the best thing to do is graph it with a calculator. Make sure it looks continuous except at x = 4, where there should be an asymptote. If we ask the calculator what the function is for x = -3, it should say "ERROR," because f(-3) is undefined.

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