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  • ️Weisstein, Eric W.
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There are several commonly used methods of defining the slippery, but extremely important, concept of a continuous function (which, depending on context, may also be called a continuous map). The space of continuous functions is denoted C^0, and corresponds to the k=0 case of a C-k function.

A continuous function can be formally defined as a function f:X->Y where the pre-image of every open set in Y is open in X. More concretely, a function f(x) in a single variable x is said to be continuous at point x_0 if

1. f(x_0) is defined, so that x_0 is in the domain of f.

2. lim_(x->x_0)f(x) exists for x in the domain of f.

3. lim_(x->x_0)f(x)=f(x_0),

where lim denotes a limit.

Many mathematicians prefer to define the continuity of a function via a so-called epsilon-delta definition of a limit. In this formalism, a limit c of function f(x) as x approaches a point x_0,

lim_(x->x_0)f(x)=c,

(1)

is defined when, given any epsilon>0, a delta>0 can be found such that for every x in some domain D and within the neighborhood of x_0 of radius delta (except possibly x_0 itself),

|f(x)-c|<epsilon.

(2)

Then if x_0 is in D and

lim_(x->x_0)f(x)=f(x_0)=c,

(3)

f(x) is said to be continuous at x_0.

If f is differentiable at point x_0, then it is also continuous at x_0. If two functions f and g are continuous at x_0, then

1. f+g is continuous at x_0.

2. f-g is continuous at x_0.

3. fg is continuous at x_0.

4. f/g is continuous at x_0 if g(x_0)!=0.

5. Providing that f is continuous at g(x_0), f degreesg is continuous at x_0, where f degreesg denotes f(g(x)), the composition of the functions f and g.

Discontinuous

The notion of continuity for a function in two variables is slightly trickier, as illustrated above by the plot of the function

z=(x^2-y^2)/(x^2+y^2).

(4)

This function is discontinuous at the origin, but has limit 0 along the line x=y, limit 1 along the x-axis, and limit -1 along the y-axis (Kaplan 1992, p. 83).

See also

C-k Function, Continuous Map, Continuously Differentiable Function, Critical Point, Differentiable, Limit, Neighborhood, Piecewise Continuous, Stationary Point Explore this topic in the MathWorld classroom

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References

Bartle, R. G. and Sherbert, D. Introduction to Real Analysis. New York: Wiley, p. 141, 1991.Kaplan, W. "Limits and Continuity." §2.4 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 82-86, 1992.

Referenced on Wolfram|Alpha

Continuous Function

Cite this as:

Weisstein, Eric W. "Continuous Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ContinuousFunction.html

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