# 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.

Discover More

## Example Sentences

I had slowly come to realize that there was a straightforward way to map the multiplication of infinite-dimensional matrices into the calculus of continuous functions.

For example, see what has happened to the idea of continuous function.

If this hypothesis were not admitted there would no longer be any way of representing the probability by a continuous function.

Then it was assumed a continuous function can change sign only by vanishing; to-day we prove it.

By a “continuous function” of one variable we always mean a function which is continuous throughout an interval.

The power series represents a continuous function in its domain of convergence (the end-points may have to be excluded).