Activity 2.10 Discontinuities

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 \(a\text{.}\))

Definition 2.10.1 One-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.

Definition 2.10.2 Removable 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.

Exercises Exercises

1.

Referring to the function \(h\) shown in the figure from Exercise 2.9.1, 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 the figure from Exercise 2.9.1, state the values of \(t\) where the function has removable discontinuities.

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