Discontinuity

Types of Discontinuity

==2024-11-17

Theory

There are many types of discontinuity. The types are jump, removable, imremovable, infinite.

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Examples

1. Continuity

x=y

2. Removable Discontinuity

Condition: Limit exists but not same as functional value. ${\lim }_{x\to x_o^{+}}f(x)={\lim }_{x\to x_0^{-}}f(x)\ne f(x_o)$

$f(x)=\{\begin{array}{ll}x& \text{if }x\ne 0\\ 1& \text{if }x=0\end{array}$,

x=y|x>0
x=y|x<0
(0,1)

$f(x)=\{\begin{array}{ll}x& \text{if }x>0\\ x& \text{if }x<0\end{array}$, No condition for 0

x=y|x>0
x=y|x<0

3. Infinite Discontinuity

Condition: Limit of the function at $x_o$ is $\infty$.

y=1/x^2

4. Jump Discontinuity

Condition: Left hand limit != Right hand limit

${\lim }_{x\to x_o^{+}}f(x)\ne {\lim }_{x\to x_0^{-}}f(x)$

y=x|x<1
y=x-1|x>=1
(1,0)

5. Oscillating Discontinuity

$y=\sin \left(\frac{1}{x}\right)$

y=\sin(1/x)

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PTR