Glossary

continuity at a point

A function $f(x)$ is continuous at a point $a$ if and only if the following three conditions are satisfied: (1) $f(a)$ is defined, (2) $\underset{x\to a}{\lim}f(x)$ exists, and (3) $\underset{x\to a}{\lim}f(x)=f(a)$

continuity from the left

A function is continuous from the left at $b$ if $\underset{x\to b^-}{\lim}f(x)=f(b)$

continuity from the right

A function is continuous from the right at $a$ if $\underset{x\to a^+}{\lim}f(x)=f(a)$

continuity over an interval

a function that can be traced with a pencil without lifting the pencil; a function is continuous over an open interval if it is continuous at every point in the interval; a function $f(x)$ is continuous over a closed interval of the form $[a,b]$ if it is continuous at every point in $(a,b)$, and it is continuous from the right at $a$ and from the left at $b$

discontinuity at a point

A function is discontinuous at a point or has a discontinuity at a point if it is not continuous at the point

infinite discontinuity

An infinite discontinuity occurs at a point $a$ if $\underset{x\to a^-}{\lim}f(x)=\pm \infty$ or $\underset{x\to a^+}{\lim}f(x)=\pm \infty$

Intermediate Value Theorem

Let $f$ be continuous over a closed bounded interval $[a,b]$; if $z$ is any real number between $f(a)$ and $f(b)$, then there is a number $c$ in $[a,b]$ satisfying $f(c)=z$

jump discontinuity

A jump discontinuity occurs at a point $a$ if $\underset{x\to a^-}{\lim}f(x)$ and $\underset{x\to a^+}{\lim}f(x)$ both exist, but $\underset{x\to a^-}{\lim}f(x) \ne \underset{x\to a^+}{\lim}f(x)$

removable discontinuity

A removable discontinuity occurs at a point $a$ if $f(x)$ is discontinuous at $a$, but $\underset{x\to a}{\lim}f(x)$ exists