2.4 Continuity  (Page 3/16)

Definition

If $f\left(x\right)$ is discontinuous at a , then

1. $f$ has a removable discontinuity at a if $\underset{x\to a}{\text{lim}}f\left(x\right)$ exists. (Note: When we state that $\underset{x\to a}{\text{lim}}f\left(x\right)$ exists, we mean that $\underset{x\to a}{\text{lim}}f\left(x\right)=L,$ where L is a real number.)
2. $f$ has a jump discontinuity at a if $\underset{x\to a^{-}}{\text{lim}}f\left(x\right)$ and $\underset{x\to a^{+}}{\text{lim}}f\left(x\right)$ both exist, but $\underset{x\to a^{-}}{\text{lim}}f\left(x\right)\ne \underset{x\to a^{+}}{\text{lim}}f\left(x\right).$ (Note: When we state that $\underset{x\to a^{-}}{\text{lim}}f\left(x\right)$ and $\underset{x\to a^{+}}{\text{lim}}f\left(x\right)$ both exist, we mean that both are real-valued and that neither take on the values ±∞.)
3. $f$ has an infinite discontinuity at a if $\underset{x\to a^{-}}{\text{lim}}f\left(x\right)=\text{±}\infty$ or $\underset{x\to a^{+}}{\text{lim}}f\left(x\right)=\text{±}\infty .$

Continuity from the right and from the left

A function $f\left(x\right)$ is said to be continuous from the right at a if $\underset{x\to a^{+}}{\text{lim}}f\left(x\right)=f\left(a\right).$

A function $f\left(x\right)$ is said to be continuous from the left at a if $\underset{x\to a^{-}}{\text{lim}}f\left(x\right)=f\left(a\right).$