Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/42

has infinitely many values for a given value of $\scriptstyle {x}$, the graph of equation$$\scriptstyle {y=\operatorname {arc~tan} x}$$consisting of infinitely many branches. If, however, we confine ourselves to any single branch, the function is continuous. For instance, if we say that $\scriptstyle {y}$ shall be the arc of smallest numerical value whose tangent is $\scriptstyle {x}$, that is, $\scriptstyle {y}$ shall take on only values between $\scriptstyle {-{\frac {\pi }{2}}}$ and $\scriptstyle {\frac {\pi }{2}}$, then we are limited to the branch passing through the origin, and the condition for continuity is satisfied.

(9) Similarly, $$\scriptstyle {\operatorname {arc~tan} {\frac {1}{x}},}$$is found to be a many-valued function. Confining ourselves to one branch of the graph of $$\scriptstyle {y=\operatorname {arc~tan} {\frac {1}{x}},}$$we see that as $\scriptstyle {x}$ approaches zero from the left, $\scriptstyle {y}$ approaches the limit $\scriptstyle {-{\frac {\pi }{2}},}$ and as $\scriptstyle {x}$ approaches zero from the right, $\scriptstyle {y}$ approaches the limit $\scriptstyle {+{\frac {\pi }{2}}.}$ Hence the function is discontinuous when $\scriptstyle {x=0.}$ Its value for $\scriptstyle {x=0}$ can be assigned at pleasure.

Functions exist which are discontinuous for every value of the independent variable within a certain range. In the ordinary applications of the Calculus, however, we deal with functions which are discontinuous (if at all) only for certain isolated values of the independent variable; such functions are therefore in general continuous, and are the only ones considered in this book.

20. Fundamental theorems on limits. In problems involving limits the use of one or more of the following theorems is usually implied. It is assumed that the limit of each variable exists and is finite.

Theorem I. The limit of the algebraic sum of a finite number of variables is equal to the like algebraic sum of the limits of the several variables.

Theorem II. The limit of the product of a finite number of variables is equal to the product of the limits of the several variables.

Theorem III. The limit of the quotient of two variables is equal to the quotient of the limits of the separate variables, provided the limit of the denominator is not zero.