Continuity of a Function

Informally, a function is continuous if you can draw its graph without lifting your pen — no breaks, jumps or holes. The precise definition pins that picture to the language of limits, and almost every exam question is really a check of one of three things at a single point.

Continuity at a point

A function $f$ is continuous at $x=c$ if all three conditions hold together:

• $f(c)$ is defined (the point exists),
• $\displaystyle\lim_{x\to c} f(x)$ exists — i.e. the left-hand limit equals the right-hand limit, $\displaystyle\lim_{x\to c^-}f(x)=\lim_{x\to c^+}f(x)$, and
• the two agree: $\displaystyle\lim_{x\to c} f(x)=f(c)$.

If even one condition fails, $f$ is discontinuous at $c$. A compact way to write the whole test is

This is the single most useful line in the chapter. For a piecewise function, the join points are exactly where the rule changes, so the standard task is to evaluate the LHL using the formula on the left, the RHL using the formula on the right, the value $f(c)$ from whichever piece owns $x=c$, and then demand all three be equal.

Continuity on an interval

$f$ is continuous on an open interval $(a,b)$ if it is continuous at every point of it. On a closed interval $[a,b]$ we additionally only require one-sided continuity at the ends: $\displaystyle\lim_{x\to a^+}f(x)=f(a)$ and $\displaystyle\lim_{x\to b^-}f(x)=f(b)$. The everyday building blocks are continuous on their natural domains:

• Every polynomial is continuous on all of $\mathbb{R}$.
• $\sin x,\ \cos x,\ e^{x}$ are continuous everywhere; $\log x$ is continuous for $x>0$.
• A rational function $\dfrac{p(x)}{q(x)}$ is continuous wherever $q(x)\ne 0$.
• $\tan x$ and $\sec x$ are continuous except where $\cos x=0$, i.e. $x=(2n+1)\dfrac{\pi}{2}$; $\cot x,\ \csc x$ except where $\sin x=0$.
• The greatest-integer function $[x]$ is discontinuous at every integer (it jumps), and $|x|$ is continuous everywhere.

Algebra of continuous functions

If $f$ and $g$ are both continuous at $c$, then so are

These rules let you certify almost any algebraic expression as continuous in a single line, without going back to limits. For example $\dfrac{x^2+1}{x-3}$ is a quotient of two polynomials (each continuous everywhere), so it is continuous wherever the denominator is non-zero — that is, for all $x\ne 3$.

Continuity of composite functions

If $g$ is continuous at $c$ and $f$ is continuous at $g(c)$, then the composite $f\circ g$ given by $(f\circ g)(x)=f\big(g(x)\big)$ is continuous at $c$. In words: a continuous function of a continuous function is continuous. This is why expressions such as $\sin(x^2)$, $e^{\cos x}$, $\sqrt{x^2+1}$ and $|\,\sin x\,|$ are continuous on their domains — each is built by composing functions already known to be continuous.

Finding unknown constants for continuity

A favourite question type gives a piecewise $f$ with one or two unknown constants and asks you to choose them so that $f$ is continuous. The recipe never changes: at each join, set LHL $=$ RHL $=f(\text{join})$ and solve the resulting equation(s). With one unknown you get one equation; with two unknowns (typically two joins or a jump plus a defined value) you get two equations to solve simultaneously.