Boundedness - Describing Functions

Mathematical Writing - Vivaldi Franco 2014

Boundedness
Describing Functions

A set $$X\subset \mathbb {R}$$ is bounded if there is an interval containing it,2 namely if

$$\begin{aligned} \exists a,b\in \mathbb {R},\,\,\forall x\in X,\,\,a<x<b \end{aligned}$$

(5.6)

or

$$\begin{aligned} \exists c\in \mathbb {R},\,\,\forall x\in X,\,\,|x|<c. \end{aligned}$$

(5.7)

The two definitions are equivalent. (Think about it.) The numbers $$a$$ and $$b$$ in (5.6) are an upper and a lower bound for $$X$$.

A real function $$f$$ is bounded if its image $$f(\mathbb {R})$$ is a bounded set. In symbols:

$$ \exists c\in \mathbb {R},\,\,\forall x\in \mathbb {R},\,\,|f(x)|< c. $$

For example, the sine function is bounded and the exponential is not. The periodic function displayed in Fig. 5.2 is bounded.

A function $$f$$ is bounded away from zero if its reciprocal is bounded. This means that for some positive constant $$c$$ we have $$|f(x)|>c$$ for all values of $$x$$. In symbols:

$$ \exists c \in \mathbb {R}^+,\,\,\forall x\in \mathbb {R},\,\,|f(x)|>c. $$

The hyperbolic cosine is bounded away from zero (what could be a value of $$c$$ in this case?) but the exponential function is not.