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continuous function

noun, Mathematics
1.
(loosely) a mathematical function such that a small change in the independent variable, or point of the domain, produces only a small change in the value of the function.
2.
(at a point in its domain) a function that has a limit equal to the value of the function at the point; a function that has the property that for any small number, a second number can be found such that when the distance between any other point in the domain and the given point is less than the second number, the difference in the functional values at the two points is less than the first number in absolute value.
3.
(at a point in a topological space) a function having the property that for any open set containing the image of the point, an open set about the given point can be found such that the image of the set is contained in the first open set.
4.
(on a set in the domain of the function or in a topological space) a function that is continuous at every point of the set.
Dictionary.com Unabridged
Based on the Random House Dictionary, © Random House, Inc. 2014.
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Examples for continuous function
  • Since pressure is a continuous function, the pressures must become equal at the wing tips.
  • Most cohesion models are defined as a continuous function of the separation distance between interacting particles.
  • Maintenance is a continuous function and a continuous responsibility of the airport sponsor.
  • The process of preparing occupational and industry projections is a continuous function.
  • Stakeholder involvement has been and will continue to be a continuous function in the identification and resolution of issues.
  • Any piece-wise continuous function with a finite integral can be accurately enough represented by a histogram.
continuous function in Technology

A function f : D -> E, where D and E are cpos, is continuous if it is monotonic and
f (lub Z) = lub f z | z in Z
for all directed sets Z in D. In other words, the image of the lub is the lub of any directed image.
All additive functions (functions which preserve all lubs) are continuous. A continuous function has a least fixed point if its domain has a least element, bottom (i.e. it is a cpo or a "pointed cpo" depending on your definition of a cpo). The least fixed point is
fix f = lub f^n bottom | n = 0..infinity
(1994-11-30)
The Free On-line Dictionary of Computing, © Denis Howe 2010 http://foldoc.org
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