4.8 Classifying discontinuities

Throughout what follows, we let $I\subseteq\mathbb{R}$ be an interval, $f\colon I\to\mathbb{R}$ and $a\in I$ . Assume $a$ is not an endpoint of $I$ . We can use one-sided limits and the idea of tending to infinity to classify different kinds of discontinuity which can occur at $a$ . First of all, we observe the following lemma, which is a direct consequence of the result of Exercise 4.72.

Lemma 4.85.

The following are equivalent:

1

$f$ is continuous at $a$ ;
2

The left limit $\displaystyle\lim_{x\to a_{-}}f(x)$ and right limit $\displaystyle\lim_{x\to a_{+}}f(x)$ both exist and satisfy

$\lim_{x\to a_{-}}f(x)=\lim_{x\to a_{+}}f(x)=f(a).$

The second characterisation of continuity in Lemma 4.85 can fail in three different ways. We use this to classify discontinuities according to three different types: removable, jump and essential discontinuities.

Removable discontinuities.

To illustrate this concept, we return to the $\mathrm{sinc}^{*}$ function first introduced in Example 4.44. In Example 4.45 we extended the definition of $\mathrm{sinc}^{*}$ to $x=0$ in such a way that the resulting function $\mathrm{sinc}$ is continuous. However, we could extend the definition of $\mathrm{sinc}^{*}$ in another way, for instance defining

(4.14) (4.14)

s\colon\mathbb{R}\to\mathbb{R};\qquad s(x):=\begin{cases}\frac{\sin x}{x}&% \text{if $x\neq 0$,}\\ \frac{1}{2}&\text{if $x=0$.}\end{cases}

However, this function is not continuous at the origin, since the choice of value $1/2$ does not ‘fill in the hole’ in the graph: see Figure 4.21. This is an example of a removable discontinuity.

Figure 4.21: The function

s\colon\mathbb{R}\to\mathbb{R}

from (4.14) has a removable discontinuity at

x=0

Definition 4.86.

We say $f$ has a removable discontinuity at $a$ if $\displaystyle\lim_{x\to a}f(x)$ exists but

\lim_{x\to a}f(x)\neq f(a).

This situation is exemplified both by the function $s$ in (4.14) and the function $\chi_{\mathbb{Z}}$ from Example 4.43: see Figure 4.21 and Figure 4.22(a).

Another way to express the definition of a removable discontinuity is that the left limit $\lim_{x\to a_{-}}f(x)$ and right limit $\lim_{x\to a_{+}}f(x)$ both exist and are equal, but their common value does not equal $f(a)$ . For instance, in the case of $s$ in (4.14), both limits are $1$ but $f(0)=0$ .

Removable discontinuities are called ‘removable’ because we can make them go away by redefining the function at $a$ . For instance, if we take the function $f$ from (4.14) and redefine the value of $f(0)$ to be $1$ instead of $1/2$ , then we go back to the function $\mathrm{sinc}$ from Example 4.45, which we know is continuous.

Jump discontinuities.

These are what we typically think of when we imagine a discontinuity.

Definition 4.87.

We say $f$ has a jump discontinuity at $a$ if $\displaystyle\lim_{x\to a_{-}}f(x)$ and $\displaystyle\lim_{x\to a_{+}}f(x)$ both exist but

\lim_{x\to a_{-}}f(x)\neq\lim_{x\to a_{+}}f(x).

This situation is exemplified by the function in Example 4.73: see Figure 4.22(b).

(a) Removable discontinuity.

(b) Jump discontinuity.

(d) Essential discontinuity due to oscillation.

Figure 4.22: Classifying discontinuities.

Essential discontinuities.

Finally, we consider the most extreme form of discontinuity, where one or both of the one-sided limits fails to exist.

Definition 4.88.

We say $f$ has a essential discontinuity at $a$ if $\displaystyle\lim_{x\to a_{-}}f(x)$ or $\displaystyle\lim_{x\to a_{+}}f(x)$ fail to exist as real numbers.

Note that the definition of essential discontinuity includes the cases where $\displaystyle\lim_{x\to a_{-}}f(x)=\pm\infty$ or $\displaystyle\lim_{x\to a_{+}}f(x)=\pm\infty$ .

Essential discontinuities can themselves be divided into two subcategories.

•

We have a vertical asymptote. This situation is exemplified by the function in Figure 4.22(c). Here $\displaystyle\lim_{x\to 0_{+}}f(x)=\infty$ : the right-hand limit does not exist as a real number, but we can still give it meaning as $\infty$ using our earlier definitions.
•

In the remaining situation, at least one of the one-sided limits fails to exist because the function oscillates wildly near $a$ . This situation is exemplified by the function in Example 4.36: see Figure 4.22(d). We provide further details below.

Example 4.89.

Let $f\colon\mathbb{R}\to\mathbb{R}$ be the function from Example 4.36, which was defined by

(4.15) (4.15)

f(x):=\begin{cases}\sin(1/x)&\text{if $x\neq 0$,}\\ 0&\text{if $x=0$.}\end{cases}

We claim that $f$ has an essential discontinuity at $a=0$ . To see this, we show that $\lim_{x\to 0_{+}}\sin(1/x)$ does not exist.

Arguing by contradiction, suppose that $\ell:=\lim_{x\to 0_{+}}\sin(1/x)\in\mathbb{R}$ exists. We can therefore apply Lemma 4.59 (we actually need a slight variant of Lemma 4.59 for one-sided limits, but the idea is the same), which says that if $(x_{n})_{n\in\mathbb{N}}$ is a sequence of real numbers satisfying $x_{n}\to 0$ as $n\to\infty$ , it follows that $\lim_{n\to\infty}\sin(1/x_{n})=\ell$ .

Let $a_{n}:=\frac{1}{2\pi n}$ for all $n\in\mathbb{N}$ , noting $a_{n}\to 0$ as $n\to\infty$ . We have $\sin(1/a_{n})=\sin(2\pi n)=0$ for all $n\in\mathbb{N}$ . Consequently,

\ell=\lim_{n\to\infty}\sin(1/a_{n})=0.

On the other hand, let $b_{n}:=\frac{2}{\pi(4n+1)}$ for all $n\in\mathbb{N}$ , noting $b_{n}\to 0$ as $n\to\infty$ . This time we have $\sin(1/b_{n})=\sin(\pi(2n+1/2))=\sin(\pi/2)=1$ for all $n\in\mathbb{N}$ . Consequently,

\ell=\lim_{n\to\infty}\sin(1/b_{n})=1.

These two observations imply $0=\ell=1$ , a contradiction.

Exercise 4.90.

Determine whether the following functions $f\colon\mathbb{R}\to\mathbb{R}$ have a removable, jump or essential discontinuity at the specified value of $a$ .

(i)

$\displaystyle f(x):=\begin{cases}\frac{5}{(x-4)^{3}},&x\neq 4,\\ 0&x=4,\end{cases}$ $a=4$ ;
(ii)

$f(x):=\begin{cases}\sin(x^{2}),&\text{$x\neq 0$,}\\ 2,&\text{$x=0$,}\end{cases}$ $a=0$ ;
(iii)

$f(x):=\begin{cases}\sin(1/x^{2}),&\text{$x\neq 0$,}\\ 0,&\text{$x=0$,}\end{cases}$ $a=0$ ;
(iv)

$f(x):=\begin{cases}x^{2},&\text{$x\leq 1$,}\\ 1-x^{3},&\text{$x>1$,}\end{cases}$ $a=1$ .