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

If temperature represents a continuous function, what kind of function would not be continuous?

From Textbooks • Dec. 1, 2021

We know that if a continuous function has a local extrema, it must occur at a critical point.

From Textbooks • Mar. 30, 2016

Let’s consider some additional situations in which a continuous function fails to be differentiable.

From Textbooks • Mar. 30, 2016

Now that we know any continuous function f defined on a closed, bounded set attains its extreme values, we need to know how to find them.

From Textbooks • Mar. 30, 2016

What is required may be expressed in mathematical language by saying that the position of a moving body must be a continuous function of the time.

From Our Knowledge of the External World as a Field for Scientific Method in Philosophy by Russell, Bertrand