Discontinuity

Definition

Removable Discontinuity

Suppose $f:\mathcal{D}\to \mathbb{R}$, and $a\in \mathcal{D}$ $\underset{x\to a^{+}}{\lim }f(x)=\underset{x\to a^{-}}{\lim }f(x)\ne f(a)$ where $a$ is called a removable discontinuity

Two one-sided limits exist and are equal, but the function value is not equal

Jump Discontinuity

Suppose $f:\mathcal{D}\to \mathbb{R}$, and $a\in \mathcal{D}$ $\underset{x\to a^{+}}{\lim }f(x)\ne \underset{x\to a^{-}}{\lim }f(x)$ where $a$ is called a jump discontinuity

Two one-sided limits exist, but not equal

Essential Discontinuity

Suppose $f:\mathcal{D}\to \mathbb{R}$, and $a\in \mathcal{D}$ $\nexists \underset{x\to a^{+}}{\lim }f(x)\vee \nexists \underset{x\to a^{-}}{\lim }f(x)$ where $a$ is called an essential discontinuity

At least one of the two one-sided limits does not exist in $\mathbb{R}$