Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/40

19. Continuity and discontinuity of functions illustrated by their graphs.

(1) Consider the function $\scriptstyle {x^{2}}$, and let
(A)	$\scriptstyle {y=x^{2}}$.

If we assume values for $\scriptstyle {x}$ and calculate the corresponding values of $\scriptstyle {y}$, we can plot a series of points. Drawing a smooth line free-hand through these points, a good representation of the general behavior of the function may be obtained. This picture or image of the function is called its graph. It is evidently the locus of all points satisfying equation (A).

Such a series or assemblage of points is also called a curve. Evidently we may assume values of $\scriptstyle {x}$ so near together as to bring the values of $\scriptstyle {y}$ (and therefore the points of the curve) as near together as we please. In other words, there are no breaks in the curve, and the function $\scriptstyle {x^{2}}$ is continuous for all values of $\scriptstyle {x}$.

(2) The graph of the continuous function $\scriptstyle {\sin x}$ is plotted by drawing the locus of

$\scriptstyle {y=\sin x}$.

It is seen that no break in the curve occurs anywhere.

(3) The continuous function $\scriptstyle {e^{x}}$ is of very frequent occurrence in the Calculus. If we plot its graph from$\scriptstyle {(e=2.718\cdots )}$$\scriptstyle {y=e^{x},}$we get a smooth curve as shown. From this it is clearly seen that,

(a) when $\scriptstyle {x=0}$, $\scriptstyle {{\underset {x=0}{\operatorname {limit} }}\;y(=e^{x})=1}$;

(b) when $\scriptstyle {x>0}$, $\scriptstyle {y(=e^{x})}$ is positive and increases as we pass towards the right from the origin;

(c) when $\scriptstyle {x<0}$, $\scriptstyle {y(=e^{x})}$ is still positive and decreases as we pass towards the left from the origin.

(4) The function $\scriptstyle {\log _{e}x}$ is closely related to the last one discussed. In fact, if we plot its graph from$\scriptstyle {y=\log _{e}x,}$it will be seen that its graph has the same relation to $\scriptstyle {OX}$ and $\scriptstyle {OY}$ as the graph of $\scriptstyle {e^{x}}$ has to $\scriptstyle {OY}$ and $\scriptstyle {OX}$.