Continuous variation

A variable $x$ is said to vary continuously through an interval $\left \lbrack a,\ b \right \rbrack$, 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 of every number between $a$ and $b$ in the order of their magnitudes. This may be illustrated geometrically as follows:

Figure 2.1: Interval from $A$ to $B$.

The origin being at $O$, layoff on the straight line the points $A$ and $B$ corresponding to the numbers $a$ and $b$. Also let the point $P$ correspond to a particular value of the variable $x$. Evidently the interval $\left \lbrack a,\ b \right \rbrack$ is represented by the segment $AB$. Now as $x$ varies continuously from $a$ to $b$ inclusive, i.e. through the interval $\left \lbrack a,\ b \right \rbrack$, the point $P$ generates the segment $AB$.