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

Lecture

Demonstration of the Theorem

Description

This lecture presents the step-by-step demonstration of a theorem related to a function G, showing the limit as h approaches 0 of G(x+h) - G(x). The proof involves the evaluation of the function at different points and the application of the mean value theorem.

In MOOCs (2)

Analysis I

Analysis I (part 7) : Indefinite and definite integrals, integration (selected chapters)

Official source

Quiz

Related concepts (20)

Function (mathematics)

In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity).

Mathematical analysis

Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).

Domain (mathematical analysis)

In mathematical analysis, a domain or region is a non-empty connected open set in a topological space, in particular any non-empty connected open subset of the real coordinate space Rn or the complex coordinate space Cn. A connected open subset of coordinate space is frequently used for the domain of a function, but in general, functions may be defined on sets that are not topological spaces.

Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus, a function defined on a subset of the real numbers with real values is called monotonic if and only if it is either entirely non-increasing, or entirely non-decreasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease.

Evaluation strategy

In a programming language, an evaluation strategy is a set of rules for evaluating expressions. The term is often used to refer to the more specific notion of a parameter-passing strategy that defines the kind of value that is passed to the function for each parameter (the binding strategy) and whether to evaluate the parameters of a function call, and if so in what order (the evaluation order). The notion of reduction strategy is distinct, although some authors conflate the two terms and the definition of each term is not widely agreed upon.

Related lectures (30)

Coq Workshop: Introduction to Interactive Theorem Proving

Introduces Coq, an interactive theorem assistant based on the Curry-Howard isomorphism.

Linear Independence: The Wronskian Concept

Explains the Wronskian and its role in determining linear independence of solutions to differential equations.

Generalized Integral: Comparison Criteria and Taylor Series

Explores series convergence criteria, generalized integrals, and Taylor series applications.

Derivatives and Local Extrema

Explores derivatives, local extrema, and function variation in mathematical analysis.

Harmonic Forms: Main Theorem