**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Lecture# Functions Composition: Continuity & Elements

Description

This lecture covers the composition of functions, continuity, and elementary functions. It explains the concept of continuity in functions, the composition of two continuous functions, and the construction of elementary functions from algebraic, exponential, logarithmic, and trigonometric functions. It also discusses the continuity of elementary functions and their domain of definition. Examples and remarks are provided to illustrate the concepts.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

In course

MATH-101(d): Analysis I

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

Related concepts (209)

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).

Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

Gamma function

In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n, Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles.

High-definition video

High-definition video (HD video) is video of higher resolution and quality than standard-definition. While there is no standardized meaning for high-definition, generally any video image with considerably more than 480 vertical scan lines (North America) or 576 vertical lines (Europe) is considered high-definition. 480 scan lines is generally the minimum even though the majority of systems greatly exceed that. Images of standard resolution captured at rates faster than normal (60 frames/second North America, 50 fps Europe), by a high-speed camera may be considered high-definition in some contexts.

High-definition television

High-definition television (HD or HDTV) describes a television system which provides a substantially higher than the previous generation of technologies. The term has been used since 1936; in more recent times, it refers to the generation following standard-definition television (SDTV), often abbreviated to HDTV or HD-TV. It is the current de facto standard video format used in most broadcasts: terrestrial broadcast television, cable television, satellite television and Blu-ray Discs.

Related lectures (1,000)

Generalized Integrals: Convergence and DivergenceMATH-101(e): Analysis I

Explores the convergence and divergence of generalized integrals using comparison methods and variable transformations.

Continuous Functions and Elementary FunctionsMATH-101(d): Analysis I

Covers the definition and properties of continuous functions on open intervals and elementary functions.

Limits and Operations on LimitsMATH-101(d): Analysis I

Covers limits, algebraic operations, and infinite limits with examples of functions' behavior near limit points.

Continuous Functions and DerivativesMATH-101(e): Analysis I

Covers the theorem of the intermediate value, derivatives, and their geometric interpretation.

Derivability and Chain RuleMATH-101(d): Analysis I

Covers the demonstration of the chain rule and the theorem of Rolle.