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

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

From Textbooks • Dec. 1, 2021

In Figure 4.31, we show that if a continuous function f has a local extremum, it must occur at a critical point, but a function may not have a local extremum at a critical point.

From Textbooks • Mar. 30, 2016

Specifically, it guarantees that any continuous function has an antiderivative.

From Textbooks • Mar. 30, 2016

As mentioned earlier, since A is a continuous function on a closed, bounded interval, by the extreme value theorem, it has a maximum and a minimum.

From Textbooks • Mar. 30, 2016

Riemann; but the failure of an attempt made by Amp�re to prove that every continuous function must be differentiable may be regarded as the first step in the theory.

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