continuous function

American  

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.

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Example Sentences

See Examples For:

continuous function

A function that has no holes or breaks in its graph is known as a continuous function.

From Textbooks ● Dec. 1, 2021

The cardinal number 𝔡 is defined as the smallest possible size of a set of continuous functions sufficient to dominate every possible continuous function.

From Scientific American ● Aug. 16, 2021

How could a continuous function avoid hitting everything between two of its values?

From Scientific American ● Feb. 17, 2019

A continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum.

From Textbooks ● Mar. 30, 2016

Differentiable Function.—The idea of the differentiation of a continuous function is that of a process for measuring the rate of growth; the increment of the function is compared with the increment of the variable.

From Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 3 "Frost" to "Fyzabad" by Various