For a function to be continuous at a point, it must be defined at that point, its limit must exist at the point, and the value of the function at that point must equal the value of the limit at that point.
Discontinuities may be classified as removable, jump, or infinite.
A function is continuous over an open interval if it is continuous at every point in the interval. It is continuous over a closed interval if it is continuous at every point in its interior and is continuous at its endpoints.
The composite function theorem states: If [latex]f(x)[/latex] is continuous at [latex]L[/latex] and [latex]\underset{x\to a}{\lim}g(x)=L[/latex], then [latex]\underset{x\to a}{\lim}f(g(x))=f(\underset{x\to a}{\lim}g(x))=f(L)[/latex].
The Intermediate Value Theorem guarantees that if a function is continuous over a closed interval, then the function takes on every value between the values at its endpoints.
Glossary
continuity at a point
A function [latex]f(x)[/latex] is continuous at a point [latex]a[/latex] if and only if the following three conditions are satisfied: (1) [latex]f(a)[/latex] is defined, (2) [latex]\underset{x\to a}{\lim}f(x)[/latex] exists, and (3) [latex]\underset{x\to a}{\lim}f(x)=f(a)[/latex]
continuity from the left
A function is continuous from the left at [latex]b[/latex] if [latex]\underset{x\to b^-}{\lim}f(x)=f(b)[/latex]
continuity from the right
A function is continuous from the right at [latex]a[/latex] if [latex]\underset{x\to a^+}{\lim}f(x)=f(a)[/latex]
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 [latex]f(x)[/latex] is continuous over a closed interval of the form [latex][a,b][/latex] if it is continuous at every point in [latex](a,b)[/latex], and it is continuous from the right at [latex]a[/latex] and from the left at [latex]b[/latex]
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 [latex]a[/latex] if [latex]\underset{x\to a^-}{\lim}f(x)=\pm \infty[/latex] or [latex]\underset{x\to a^+}{\lim}f(x)=\pm \infty[/latex]
Intermediate Value Theorem
Let [latex]f[/latex] be continuous over a closed bounded interval [latex][a,b][/latex]; if [latex]z[/latex] is any real number between [latex]f(a)[/latex] and [latex]f(b)[/latex], then there is a number [latex]c[/latex] in [latex][a,b][/latex] satisfying [latex]f(c)=z[/latex]
jump discontinuity
A jump discontinuity occurs at a point [latex]a[/latex] if [latex]\underset{x\to a^-}{\lim}f(x)[/latex] and [latex]\underset{x\to a^+}{\lim}f(x)[/latex] both exist, but [latex]\underset{x\to a^-}{\lim}f(x) \ne \underset{x\to a^+}{\lim}f(x)[/latex]
removable discontinuity
A removable discontinuity occurs at a point [latex]a[/latex] if [latex]f(x)[/latex] is discontinuous at [latex]a[/latex], but [latex]\underset{x\to a}{\lim}f(x)[/latex] exists
