Continuous

Pronunciation: /kənˈtɪn yu əs/ Explain
Click on the point on the black slider and drag it to see various functions. Click on the blue point on the x-axis and drag it to see the values of the functions for the values of x.

All of the discontinuous functions shown have a discontinuity at x=1.
Manipulative 1 - Discontinuous Functions Created with GeoGebra.

A function or curve is continuous if it is an unbroken curve. A more rigorous definition of continuity is:

A function is continuous at x if f(x) is defined and the limit of f(a) as a approaches x from both sides = f(x).

A function is continuous on a subdomain if it is continuous at all points in the subdomain. A function is continuous on the entire domain if it is continuous at all points in the domain. A function is discontinuous if it is not continuous. Click on the point on the slider in manipulative 1 and drag it to see various continuous and discontinuous functions.

There is a distinction between continuous on the left and continuous on the right. Take the ceiling function at x=1 as an example. At x=1, the ceiling function is continuous on the left, as the limit of f(a) as a approaches 1 from the left is 1. However, the limit of f(a) as a approaches 1 from the right is 2. To be continuous at x=1, f(x) must be continuous from the left and continuous from the right.

Continuity is important in analyzing functions, particularly in calculus. In calculus, one finds the derivative of a function. The derivative of a function exists only on subdomains that are continuous.

Cite this article as:

McAdams, David E. Continuous. 6/25/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/continuous.html.

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Revision History

6/25/2018: Removed broken links, updated license, implemented new markup, updated GeoGebra apps. (McAdams, David E.)

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