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 key advantage of the fundamental groupoid construction is that it is “functorial,” meaning that a continuous function f: X → Y between topological spaces gives rise to a corresponding transformation π1f: π1X → π1Y between the fundamental groupoids.

From Scientific American

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

A variant of this definition gives the cardinal number 𝔟, namely the smallest size of a family B with the property that there is no continuous function that dominates all functions of B. It can be shown that ℵ1 ≤ 𝔟 ≤ 𝔡 ≤ 2ℵ0 holds.

From Scientific American

Weierstrass wanted to know whether there was a limit to how not differentiable a continuous function could be, and this example shows that it can be pretty darn non-differentiable!

From Scientific American

In rough terms, when you think about graphs of functions, a continuous function is one that doesn’t have jumps, and a differentiable function is one that doesn’t have corners or spikes.

From Scientific American