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

Lecture

Properties of Continuous Functions: Maximum and Minimum

In course

MATH-101(g): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable. ...

Description

This lecture discusses the properties of continuous functions, focusing on the existence of maximum and minimum values. It explores the concept of globleaux and theorems related to intervals. The instructor explains how to determine maximum and minimum values and the existence of intermediate values between them.

Instructors (2)

Annalisa Buffa

Lénaïc Chizat

Official source

Related concepts (32)

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 .

Nowhere continuous function

In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If is a function from real numbers to real numbers, then is nowhere continuous if for each point there is some such that for every we can find a point such that and . Therefore, no matter how close we get to any fixed point, there are even closer points at which the function takes not-nearby values.

Interval estimation

In statistics, interval estimation is the use of sample data to estimate an interval of possible values of a parameter of interest. This is in contrast to point estimation, which gives a single value. The most prevalent forms of interval estimation are confidence intervals (a frequentist method) and credible intervals (a Bayesian method); less common forms include likelihood intervals and fiducial intervals.

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 .

Credible interval

In Bayesian statistics, a credible interval is an interval within which an unobserved parameter value falls with a particular probability. It is an interval in the domain of a posterior probability distribution or a predictive distribution. The generalisation to multivariate problems is the credible region. Credible intervals are analogous to confidence intervals and confidence regions in frequentist statistics, although they differ on a philosophical basis: Bayesian intervals treat their bounds as fixed and the estimated parameter as a random variable, whereas frequentist confidence intervals treat their bounds as random variables and the parameter as a fixed value.

Related lectures (34)

Continuous Functions: Definitions and Criteria

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

Darboux Theorem: Advanced Analysis I

Explores the Darboux theorem for continuous functions on closed intervals, emphasizing uniform continuity and function behavior implications.

The Intermediate Value Theorem MOOC: Analysis I MOOC: Analysis I (part 5): Continuous and derivable functions, the derivative function

Explains the Intermediate Value Theorem for continuous functions on closed intervals.

Max Local Exercise

Covers finding unique local maxima and functions oscillating infinitely.

Fundamental Theorem of Analysis: Integral (Part 2)