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Discontinuity

Determine the discontinuity at the given point and identify it as infinite, jump, or removable.

Infinite discontinuity – When a function increases or decreases indefinitely at a given point.

Example: $f(x)=\frac{1}{x^2},\ x=0\rightarrow f(0)=\frac{1}{(0)^2}=\frac{1}{0}\ undefined$

x	-0.01	-0.001	0	0.001	0.01
f(x)	$1\times 10^{4}$	$1\times 10^{6}$	–	$1\times 10^{4}$	$1\times 10^{4}$

Infinite

Jump discontinuity – At a given point the function on the left and the right have two distinct values.

Example:

$f(x)=x^{2}-1\ if\ x>-2$

$f(x)=x^{2}-1\ if\ x\leq -2,\ x=-2$

$f(-2)=(-2)^{2}-1=4-1=3$

$f(-2)=(-2)-5=-7$

x	-2.1	-2.01	-2	-1.99	-1.9
f(x)	-7.1	-7.01	–	2.96	2.61

Jump

Removable discontinuity – The function is continuous except for a hole at the given point.

Example:

$f(x)=\frac{x+3}{x^{2}-9},\ x=-3\ and\ x=3$

$f(-3)=\frac{-3+3}{(-3)^{2}-9}=\frac{0}{0}$

$f(3)=\frac{3+3}{(3)^{2}-9}=\frac{6}{0}$

x	-3.01	-3.001	-3	-2.999	-2.99
f(x)	-0.166	-0.167	–	-0.167	-0.167

Removable, $-\frac{1}{6}$

x	2.99	2.999	3	3.001	3.01
f(x)	-100	-1000	–	1000	100

Infinite