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Some of the content of this guide was modeled after a guide originally created by Openstax and has been adapted for the GPRC Learning Commons in October 2021. The graphs are generated using Desmos. This work is licensed under a Creative Commons BY 4.0 International License.


A function f(x) is continuous at a number a if

This definition requires three statements to hold true for f(x) to be continuous at a:

  1. f(a) is defined, or a is in the domain of f(x)
  2. , f(x) must be defined on an open interval that contains a:

Example 1: Discontinuous at a point

This function is discontinuous at a = 3 since f(3) is not defined.

Example 2: Discontinuous at the limit

This function is discontinuous at a = 3 since the limit of the function is undefined.

Example 3: Discontinuous at the function

This function has a discontinuity at a = 3 since the limit of the function and the value of the function are not equal.

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