Continuity

Continuity is a property of a function whose values approach its value at a point as inputs approach that point. On a domain, it means this condition holds at every point.

Limit of a function: A limit is the value that a function’s outputs approach as its input approaches a specified point. Continuity at a point is defined by the function’s limit there matching its value.
Real number: A member of the complete ordered number system used to model quantities on a continuous number line. Real-valued functions provide the standard setting for elementary continuity.
Extreme value theorem: A theorem stating that a continuous real-valued function on a closed bounded interval attains a maximum and a minimum. Continuity guarantees extrema when the domain is compact.
Differentiability: Differentiability is the property of having a well-defined linear rate of change at a point. Differentiability implies continuity, but continuous functions need not be differentiable.
Dirichlet function: The function on the real numbers equal to one on rationals and zero on irrationals. It is discontinuous at every real number, showing how densely interwoven inputs can defeat continuity.
Epsilon–delta definition: A formal definition of limits using bounds on input and output distances. It states exactly how nearby inputs must keep outputs close to the function’s value.
Open set: A set is open when each of its points has a neighborhood contained within the set. Neighborhoods express how inputs can vary around a point inside a domain.
Weierstrass approximation theorem: A theorem stating that every continuous function on a closed bounded interval can be uniformly approximated by polynomials. Continuity is precisely the condition that permits uniform polynomial approximation on such intervals.
Lipschitz continuity: A function is Lipschitz continuous when output distances are bounded by a constant times input distances. This stronger condition supplies a quantitative bound absent from ordinary continuity.
Thomae's function: A function equal to zero on irrationals and to the reciprocal of a reduced denominator on rationals. It is continuous at irrationals but discontinuous at every rational, illustrating point-dependent behavior.

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