Discontinuous 2003-07-14 13:11:17 2006-10-30 09:47:47 Definition continuous removable discontinuity saltus jump jump discontinuity discontinuity of the second kind discontinuity of the first kind essential discontinuity % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\sR}[0]{\mathbb{R}} \newcommand{\sC}[0]{\mathbb{C}} \newcommand{\sN}[0]{\mathbb{N}} \newcommand{\sZ}[0]{\mathbb{Z}} % The below lines should work as the command % \renewcommand{\bibname}{References} % without creating havoc when rendering an entry in % the page-image mode. \makeatletter \@ifundefined{bibname}{}{\renewcommand{\bibname}{References}} \makeatother \newcommand*{\norm}[1]{\lVert #1 \rVert} \newcommand*{\abs}[1]{| #1 |} \PMlinkescapeword{formula} \PMlinkescapeword{types} \PMlinkescapeword{satisfies} \PMlinkescapeword{simple} \section*{Definition} Suppose $A$ is an open set in $\sR$ (say an interval $A=(a,b)$, or $A=\sR$), and $f:A\to \sR$ is a function. Then $f$ is \emph{discontinuous} at $x\in A$, if $f$ is not continuous at $x$. One also says that $f$ is discontinuous at all boundary points of $A$. We know that $f$ is continuous at $x$ if and only if $\lim_{z\to x} f(z)=f(x)$. Thus, from the properties of the one-sided limits, which we denote by $f(x+)$ and $f(x-)$, it follows that $f$ is discontinuous at $x$ if and only if $f(x+)\neq f(x)$, or $f(x-)\neq f(x)$. If $f$ is discontinuous at $x\in\overline{A}$, the closure of $A$, we can then distinguish four types of different discontinuities as follows \cite{hoskins, contline}: \begin{enumerate} \item If $f(x+)=f(x-)$, but $f(x)\neq f(x\pm)$, then $x$ is called a \emph{removable discontinuity} of $f$. If we modify the value of $f$ at $x$ to $f(x)=f(x\pm)$, then $f$ will become continuous at $x$. This is clear since the modified $f$ (call it $f_0$) satisfies $f_0(x) = f_0(x+)=f_0(x-).$ \item If $f(x+)=f(x-)$, but $x$ is not in $A$ (so $f(x)$ is not defined), then $x$ is also called a \emph{removable discontinuity}. If we assign $f(x)=f(x\pm)$, then this modification renders $f$ continuous at $x$. \item If $f(x-)\neq f(x+)$, then $f$ has a \emph{jump discontinuity} at $x$ Then the number $f(x+)-f(x-)$ is then called the \emph{jump}, or \emph{saltus}, of $f$ at $x$. \item If either (or both) of $f(x+)$ or $f(x-)$ does not exist, then $f$ has an \emph{essential discontinuity} at $x$ (or a \emph{discontinuity of the second kind}). \end{enumerate} Note that $f$ may be continuous (continuous in all points in $A$), but still have discontinuities in $\overline{A}$ \section*{ Examples} \begin{enumerate} \item Consider the function $f:\sR\to \sR$ given by \[ f(x)=\begin{cases} 1 & \text{when }x\neq 0, \\ 0 & \text{when }x=0. \end{cases} \] Since $f(0-)=1$, $f(0)=0$, and $f(0+)=1$, it follows that $f$ has a removable discontinuity at $x=0$. If we modify $f(0)$ so that $f(0)=1$, then $f$ becomes the continuous function $f(x)=1$. \item Let us consider the function defined by the formula \[ f(x) = \frac{\sin x }{x} \] where $x$ is a nonzero real number. When $x=0$, the formula is undefined, so $f$ is only determined for $x\neq 0$. Let us show that this point is a removable discontinuity. Indeed, it is easy to see that $f$ is continuous for all $x\neq 0$, and using \PMlinkname{L'H\^opital's rule}{LHpitalsRule} we have $f(0+)=f(0-)=1$. Thus, if we assign $f(0)=1$, then $f$ becomes a continuous function defined for all real $x$. In fact, $f$ can be made into an analytic function on the whole complex plane. \newcommand{\signum}[0]{\mathop{\mathrm{sign}}} \item The signum function $\signum\colon\sR\to \sR$ is defined as \[ \signum (x) =\begin{cases} -1 & \text{when }x<0, \\ 0 & \text{when } x=0, \text{ and}\\ 1 & \text{when } x>0. \end{cases} \] Since $\signum(0+)=1$, $\signum(0)=0$, and since $\signum(0-)=-1$, it follows that $\signum$ has a jump discontinuity at $x=0$ with jump $\signum(0+)-\signum(0-)=2$. \item The function $f:\sR\to\sR$ given by \[ f(x) =\begin{cases} 1 & \text{when }x= 0, \\ \sin(1/x) & \text{when } x\neq 0 \end{cases} \] has an essential discontinuity at $x=0$. See \cite{contline} for details. \end{enumerate} \section*{General Definition} Let $X,Y$ be topological spaces, and let $f$ be a mapping $f:X\to Y$. Then $f$ is \emph{discontinuous} at $x\in X$, if $f$ is not \PMlinkname{continuous at}{Continuous} $x$. In this generality, one generally does not classify discontinuities quite so closely, since they can have quite complicated behaviour. \section*{Notes} A jump discontinuity is also called a \emph{simple discontinuity}, or a \emph{ discontinuity of the first kind}. An \emph{essential discontinuity} is also called a \emph{discontinuity of the second kind}. \begin{thebibliography}{9} \bibitem{hoskins} R.F. Hoskins, \emph{Generalised functions}, Ellis Horwood Series: Mathematics and its applications, John Wiley \& Sons, 1979. \bibitem{contline} P. B. Laval, \PMlinkexternal{http://science.kennesaw.edu/~plaval/spring2003/m4400_02/Math4400/contwork.pdf}{http://science.kennesaw.edu/~plaval/spring2003/m4400_02/Math4400/contwork.pdf}. \end{thebibliography}