Continuous function

A function is continuous when sufficiently small changes in its input produce arbitrarily small changes in its output. At a point, this is equivalent to the function’s limit there equaling its value.

Epsilon–delta definition: A quantified definition of limits and continuity using arbitrarily small positive tolerances. It states exactly how input tolerance guarantees output tolerance at a point.
Open set: A set is open when each of its points has a neighborhood contained within the set. Open sets let topology define continuity without measuring distances.
Intermediate value theorem: A theorem stating that a continuous real-valued function on an interval takes every intermediate value between its endpoint values. Continuity prevents a function from skipping values on an interval.
Differentiable function: A function is differentiable at a point when its derivative exists there as a finite limit of difference quotients. Differentiability implies continuity at a point, but continuity alone does not imply differentiability.
Limit of a function: The value that a function’s outputs approach as its input approaches a specified point. Continuity at a point means this limiting value equals the function’s value there.
Neighborhood (mathematics): A neighborhood of a point is a set containing an open set that includes that point. Continuity can be described by mapping sufficiently small neighborhoods into prescribed neighborhoods.
Extreme value theorem: A theorem stating that a continuous real-valued function on a compact space attains a maximum and minimum. It shows how continuity and compactness guarantee actual extrema.
Lipschitz continuity: A function is Lipschitz continuous when output differences are bounded by a constant times input distance. This stronger condition gives a fixed quantitative bound on changes in output.
Continuity at a point: A property requiring a function’s limit at a point to equal its value there. It is the local form of continuity, expressed directly through limits.
Metric space: A set equipped with a distance function satisfying the axioms of a metric. Distances in metric spaces make the epsilon–delta definition possible.

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