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Continuous Function

Basic definition

A continuous function is function with no jumps, gaps, or undefined points.

Detailed definition

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.

Related terms

C^k function | continuously differentiable function | continuous map | critical point | differentiable | limit | neighborhood | piecewise continuous | stationary point

Educational grade level

college level (AP calculus AB)

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