When a function has a discontinuity at $a\text{,}$ the function is sometimes continuous from only the right or only the left at $a\text{.}$ (Please note that when we say “the function is continuous at $a$” we mean that the function is continuous from both the right and left at $a\text{.}$)

##### Definition2.10.1One-sided Continuity

The function $f$ is continuous from the left at $a$ if and only if

\begin{equation*} \lim\limits_{x\to a^{-}}\fe{f}{x}=\fe{f}{a}\text{.} \end{equation*}

The function $f$ is continuous from the right at $a$ if and only if

\begin{equation*} \lim\limits_{x\to a^{+}}\fe{f}{x}=\fe{f}{a}\text{.} \end{equation*}

Some discontinuities are classified as removable discontinuities. Discontinuities that are holes or skips (holes with a secondary point) are removable.

##### Definition2.10.2Removable Discontinuity

We say that $f$ has a removable discontinuity at $a$ if $f$ is discontinuous at $a$ but $\lim\limits_{x\to a}\fe{f}{x}$ exists.

# Subsection2.10.1Exercises

##### 1

Referring to the function $h$ shown in Figure 2.9.2, state the values of $t$ where the function is continuous from the right but not the left. Then state the values of $t$ where the function is continuous from the left but not the right.

##### 2

Referring again to the function $h$ shown in Figure 2.9.2, state the values of $t$ where the function has removable discontinuities.