\chapter*{CHAPTER~II}

\section*{VARIABLES AND FUNCTIONS}

\paragraph{6.} \textbf{Variables and constants.} A \emph{variable} is a quantity to which an
unlimited number of values can be assigned. Variables are denoted
by the later letters of the alphabet. Thus, in the equation of a
straight line,
\[
\frac{x}{a} + \frac{y}{a} =1,
\]
$x$ and~$y$ may be considered as the variable coördinates of a point
moving along the line.

A quantity whose value remains unchanged is called a \emph{constant}.

\emph{Numerical} or \emph{absolute constants} retain the same values in all problems,
as $2$, $5$, $\sqrt{7}$, $\pi$, etc.

\emph{Arbitrary constants}, or \emph{parameters}, are constants to which any one
of an unlimited set of numerical values may be assigned, and they
are supposed to have these assigned values throughout the investigation.
They are usually denoted by the earlier letters of the
alphabet. Thus, for every pair of values arbitrarily assigned to $a$
and~$b$, the equation
\[
\frac{x}{a} + \frac{y}{b} = 1
\]
represents some particular straight line.

\paragraph{7.} \textbf{Interval of a variable.} Very often we confine ourselves to a
portion only of the number system. For example, we may restrict
our variable so that it shall take on only such values as lie between
$a$ and~$b$, where $a$ and~$b$ may be included, or either or both excluded.
We shall employ the symbol $[a, b]$, $a$ being less than~$b$, to represent
the numbers $a$, $b$, and all the numbers between them, unless otherwise
stated. This symbol $[a, b]$ is read \emph{the interval from~$a$ to~$b$}.

\paragraph{8.} \textbf{Continuous variation.} A variable $x$ is said to vary continuously
through an interval $[a, b]$, % [** F1: Unnecessary comma?]
when $x$ starts with the value~$a$ and increases
until it takes on the value~$b$ in such a manner as to assume the value
