Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Continuous Functions and Continuity Extension

Description

This lecture covers the concept of continuous functions and continuity extension, defining continuous functions on a domain and their extension to the closure of the domain. It explores the conditions for a function to be continuous at a point and provides examples of continuity extensions.

In MOOCs (2)

Analysis I

Analysis I (part 4) : Limit of a function, continuous functions

Official source

Quiz

Related concepts (20)

Continuous function

In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is .

Uniform continuity

In mathematics, a real function of real numbers is said to be uniformly continuous if there is a positive real number such that function values over any function domain interval of the size are as close to each other as we want. In other words, for a uniformly continuous real function of real numbers, if we want function value differences to be less than any positive real number , then there is a positive real number such that at any and in any function interval of the size .

Modulus of continuity

In mathematical analysis, a modulus of continuity is a function ω : [0, ∞] → [0, ∞] used to measure quantitatively the uniform continuity of functions. So, a function f : I → R admits ω as a modulus of continuity if and only if for all x and y in the domain of f. Since moduli of continuity are required to be infinitesimal at 0, a function turns out to be uniformly continuous if and only if it admits a modulus of continuity. Moreover, relevance to the notion is given by the fact that sets of functions sharing the same modulus of continuity are exactly equicontinuous families.

Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function (and is related to the modulus of uniform continuity).

Closure (mathematics)

In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are. Similarly, a subset is said to be closed under a collection of operations if it is closed under each of the operations individually. The closure of a subset is the result of a closure operator applied to the subset.

Related lectures (50)

Continuous Functions: Definitions and Criteria

Covers the definition and criteria for continuous functions and explores the intermediate value theorem.

Continuous Functions: Definitions and Continuity Criteria

Explains continuous functions, criteria for continuity, and continuity concepts at points and intervals.

Continuous Functions: Theory and Applications

Explores continuous functions, Cauchy criterion, and function extension by continuity.

Continuous Functions on Closed Interval MOOC: Analysis I MOOC: Analysis I (part 5): Continuous and derivable functions, the derivative function

Explores continuous functions on closed intervals, emphasizing the importance of understanding definitions for continuity.

Continuous Functions: Criteria and Equivalences