**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# Continuous Functions: Definitions and Criteria

Description

This lecture covers the definition and criteria for continuous functions, including the three conditions for continuity at a point, the Cauchy criterion for continuous functions, and the extension of functions by continuity. It also explores functions continuous on an interval, the intermediate value theorem, and the corollary related to the existence of solutions for specific equations.

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

Instructor

Related concepts (112)

Related lectures (433)

MATH-101(e): Analysis I

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

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 .

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 .

In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants C ≥ 0, α > 0, such that for all x and y in the domain of f. More generally, the condition can be formulated for functions between any two metric spaces. The number α is called the exponent of the Hölder condition. A function on an interval satisfying the condition with α > 1 is constant. If α = 1, then the function satisfies a Lipschitz condition.

In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory. The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function.

The exponential function is a mathematical function denoted by or (where the argument x is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers.

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

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

Derivatives and FunctionsMATH-101(e): Analysis I

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

Limits and Continuity

Covers limits, continuity, and the intermediate value theorem in functions.

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

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

Functions Composition: Continuity & ElementsMATH-101(d): Analysis I

Covers the composition of functions, continuity, and elementary functions, explaining the concept of continuity and the construction of elementary functions.