4.2 Functions on the real line: basic properties

We turn to examining some basic behaviours of functions defined on the real line. This discussion mirrors (and in fact, generalises) the discussion on sequences in Section 2.3.

Monotone functions

We first consider functions whose values either get larger or smaller as $x$ increases.

Definition 4.9.

Let $E\subseteq\mathbb{R}$ and $X\subseteq E$ be nonempty and $f\colon E\to\mathbb{R}$ be a function.

1

We say $f$ is increasing on $X$ if $f(x)<f(y)$ for all $x$ , $y\in X$ with $x<y$ .
2

We say $f$ is decreasing on $X$ if $f(x)>f(y)$ for all $x$ , $y\in X$ with $x<y$ .
3

We say $f$ is strictly monotone on $X$ if it is either increasing or decreasing on $X$ .

Sometimes we say strictly increasing or strictly decreasing on $X$ , to emphasise that the inequalities are strict (as opposed to $\leq$ or $\geq$ ). If $X=E$ is the whole domain, then we often drop ‘on $X$ ’ and simply say $f$ is increasing, decreasing or strictly monotone.

Example 4.10.

We consider some familiar examples.

1.

From the definition in terms of the unit circle (Figure 4.4), we can see that the function $\sin\colon\mathbb{R}\to\mathbb{R}$ is increasing on $[-\pi/2,\pi/2]$ and decreasing on $[\pi/2,3\pi/2]$ . However $\sin\colon\mathbb{R}\to\mathbb{R}$ is neither increasing nor decreasing on $[-\pi/2,3\pi/2]$ .
2.

The function $\exp\colon\mathbb{R}\to\mathbb{R}$ is increasing.

As noted earlier, $\exp(y)>\exp(x)$ for all $y>x\geq 0$ . This is also true for all $x,y\in\mathbb{R}$ , as we will see later in the course.

Exercise 4.11.

Consider the function $f\colon[0,\infty)\to\mathbb{R}$ given by $f(x):=1-1/(x^{2}+1)$ for all $x\geq 0$ . Sketch the graph of $f$ and prove that it is increasing.

Exercise 4.12.

Show that $\log\colon(0,\infty)\to\mathbb{R}$ is increasing. Hint: use the fact from Example 4.10 that $\exp$ is increasing, and that $\log$ is the inverse of $\exp$ .

Remark 4.13.

Recall from Definition 2.1 that a sequence is a function $a\colon\mathbb{N}\to\mathbb{R}$ . Because of this, the above definition generalises our existing definitions of increasing and decreasing sequences (if we set $E=X=\mathbb{N}$ , we recover our earlier definitions). Similar remarks also apply to the forthcoming definitions, which generalise established concepts for sequences.

Definition 4.14.

Let $E\subseteq\mathbb{R}$ and $X\subseteq E$ be nonempty and $f\colon E\to\mathbb{R}$ be a function.

1

We say $f$ is non-decreasing on $X$ if $f(x)\leq f(y)$ for all $x$ , $y\in X$ with $x\leq y$ ;
2

We say $f$ is non-increasing on $X$ if $f(x)\geq f(y)$ for all $x$ , $y\in X$ with $x\leq y$ ;
3

We say $f$ is monotone on $X$ if it is either non-decreasing or non-decreasing on $X$ .

As before, if $X=E$ is the whole domain, then we often drop ‘on $X$ ’ and simply say $f$ is non-decreasing, non-increasing or monotone.

Example 4.15.

The function $p_{2}\colon\mathbb{R}\to\mathbb{R}$ given by $p_{2}(x):=x^{2}$ is strictly monotone on $[-1,0]$ or $[0,1]$ but it is not monotone on $[-1,1]$ .

•

For $x,y\in[-1,0]$ , $x<y\implies x^{2}>y^{2}$ .
•

For $x,y\in[0,1]$ , $x<y\implies x^{2}<y^{2}$ .
•

Note that $p_{2}(-1)=1>0=p_{2}(0)$ while $p_{2}(0)=0<1=p_{2}(1)$ .

Exercise 4.16.

Let $\chi_{\mathbb{Q}}\colon\mathbb{R}\to\mathbb{R}$ be the Dirichlet function from Example 4.3. Show that for any $a$ , $b\in\mathbb{R}$ with $a<b$ , the function $\chi_{\mathbb{Q}}$ is not monotone on the interval $(a,b)$ .

Bounded and unbounded functions

The functions $x\mapsto 1-1/(x^{2}+1)$ and $x\mapsto x^{2}$ are both increasing on $[0,\infty)$ , but exhibit very different behaviours.

Definition 4.17.

Let $E\subseteq\mathbb{R}$ and $X\subseteq E$ be nonempty and $f\colon E\to\mathbb{R}$ be a function.

1

We say $f$ is bounded above on $X$ if there exists some $M\in\mathbb{R}$ such that $f(x)\leq M$ for all $x\in X$ .
2

We say $f$ is bounded below on $X$ if there exists some $M\in\mathbb{R}$ such that $f(x)\geq M$ for all $x\in X$ .
3

We say $f$ is bounded on $X$ if it is both bounded above and bounded below on $X$ . Otherwise, $f$ is unbounded on $X$ .

Once again, if $X=E$ is the whole domain, then we often drop ‘on $X$ ’ and simply say $f$ is bounded above, bounded below, bounded or unbounded.

Note that, by definition, the function $f\colon E\to\mathbb{R}$ is bounded above (respectively, below) if and only the set $\{f(x):x\in E\}$ is bounded above (respectively, below) in the sense introduced in Definition 1.1 (respectively, Definition 1.39).

Example 4.18.

We return to the examples mentioned above.

1.

The function $f\colon[0,\infty)\to\mathbb{R}$ given by $f(x):=1-1/(x^{2}+1)$ for all $x\geq 0$ satisfies $0\leq f(x)\leq 1$ for all $x\geq 0$ and so it is bounded. This gives an example of a function which is increasing and bounded above.
2.

The function $p_{2}\colon[0,\infty)\to\mathbb{R}$ given by $p_{2}(x):=x^{2}$ is bounded below since $p_{2}(x)\geq 0$ for all $x\geq 0$ . However, it is not bounded above. Indeed, given any $M>0$ , if we choose $x_{0}:=M+1$ , then $p_{2}(x_{0})=(M+1)^{2}>M$ , so $M$ is not an upper bound for $p_{2}$ . Since $M$ was arbitrary, there does not exist an upper bound for the function.

Exercise 4.19.

Consider the reciprocal function $r\colon\mathbb{R}\setminus\{0\}\to\mathbb{R}$ given by $r(x):=1/x$ . Let $\delta>0$ .

(i)

Show that $r$ is bounded on $(-\infty,-\delta]\cup[\delta,\infty)$ .
(ii)

Show that $r$ is unbounded on $(-\delta,\delta)\setminus\{0\}$ .