2.3 – Continuity

Continuity at a Point

A function is continuous if it can be drawn in one motion without lifting the pen.

$y=f(x)$ is continuous at an interior point $x=c$ of its domain if ${\lim }_{x\to c}f(x)=f(c)$.

$y=f(x)$ is continuous at an endpoint $x=b$ of its domain if ${\lim }_{x\to b^{\pm }}f(x)=f(b)$.

Interval Notation

Open: $(a,b)=a<x<b$

Closed: $[a,b]=a\le x\le b$

Infinity is always exclusive: $\text{ARN}\to (-\infty ,\infty )$

Discontinuity

Jump discontinuity: ${\lim }_{x\to b^{-}}f(x)\ne {\lim }_{x\to b^{+}}f(x)$

Removable discontinuity: ${\lim }_{x\to b}f(x)\ne f(b)$

• Removable discontinuities can be removed by setting $f(b)={\lim }_{x\to b}f(x)$

Infinite discontinuity: ${\lim }_{x\to b}f(x)=\pm \infty$

Oscillating discontinuity: $f(x)$ oscillates as it approaches the limit point

• E.g., ${\lim }_{x\to 0}\sin \frac{1}{x}$

Combinations

Given that $f$ and $g$ are continuous functions at $x=c$:

$f+g$

$f-g$

$f\cdot g$

$k\cdot g$, for any number $k$

$\frac{f}{g}$, provided $g(c)\ne 0$

are all continuous.

Composites

Given that $f(x)$ and $g(x)$ are continuous at $x=c$, the composite $f(g(x))$ is continuous at $c$.

The Intermediate Value Theorem

Given that the function $y=f(x)$ is continuous on the closed interval $[a,b]$, all values between $f(a)$ and f(b) exist.