Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

6: Continuity - What It Isn’t and What It Is

  • Page ID
    7957
  • [ "article:topic-guide", "authorname:eboman" ]

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    • 6.1: An Analytic Definition of Continuity
      Before the invention of calculus, the notion of continuity was treated intuitively if it was treated at all. At first pass, it seems a very simple idea based solidly in our experience of the real world. Standing on the bank we see a river flow past us continuously, not by tiny jerks. Even when the flow might seem at first to be discontinuous, as when it drops precipitously over a cliff, a closer examination shows that it really is not. As the water approaches the cliff it speeds up.
    • 6.2: Sequences and Continuity
      We will examine an alternative way to prove that the function is not continuous at a≠0 by looking at the relationship between our definitions of convergence and continuity. The two ideas are actually quite closely connected, as illustrated by the following very useful theorem.
    • 6.3: The Definition of the Limit of a Function
      Since these days the limit concept is generally regarded as the starting point for calculus, you might think it is a little strange that we’ve chosen to talk about continuity first. But historically, the formal definition of a limit came after the formal definition of continuity. In some ways, the limit concept was part of a unification of all the ideas of calculus that were studied previously and, subsequently, it became the basis for all ideas in calculus.
    • 6.4: The Derivative - An Afterthought
      Along with the integral, the derivative is one of the most powerful and useful mathematical objects ever devised and we’ve been working very hard to provide a solid, rigorous foundation for it.  On the other hand, now that we have built up all of the machinery we need to define and explore the concept of the derivative it will appear rather pedestrian alongside ideas like the convergence of power series, Fourier series, and the bizarre properties of Q and R .
    • 6.E: Continuity - What It Isn’t and What It Is (Exercises)

    Thumbnail: Cauchy around 1840. Lithography by Zéphirin Belliard after a painting by Jean Roller. Image used with permission (Public Domain).

    Contributors

    • Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia)