Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/39

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THEORY OF LIMITS

15

Case I. As an example illustrating a simple case of a function continuous for a particular value of the variable, consider the function

$\scriptstyle {f(x)={\frac {x^{2}-4}{x-2}}}$ .

For $\scriptstyle {x=1}$ , $\scriptstyle {f(x)=f(1)=3}$ . Moreover, if $\scriptstyle {x}$ approaches the limit $\scriptstyle {1}$ in any manner, the function $\scriptstyle {f(x)}$ approaches $\scriptstyle {3}$ as a limit. Hence the function is continuous for $\scriptstyle {x=1}$ .

Case II. The definition of a continuous function assumes that the function is already defined for $\scriptstyle {x=a}$ . If this is not the case, however, it is sometimes possible to assign such a value to the function for $\scriptstyle {x=a}$ that the condition of continuity shall be satisfied. The following theorem covers these cases.

Theorem. If $\scriptstyle {f(x)}$ is not defined for $\scriptstyle {x=a}$ , and if

$\scriptstyle {{\underset {x=a}{\operatorname {limit} }}\;f(x)=B}$ ,

then $\scriptstyle {f(x)}$ will be continuous for $\scriptstyle {x=a}$ , if $\scriptstyle {B}$ is assumed as the value of $\scriptstyle {f(x)}$ for $\scriptstyle {x=a}$ . Thus the function

$\scriptstyle {\frac {x^{2}-4}{x-2}}$

is not defined for $\scriptstyle {x=2}$ (since then there would be division by zero). But for every other value of $\scriptstyle {x}$ ,

$\scriptstyle {{\frac {x^{2}-4}{x-2}}=x+2}$ ;

and

\scriptstyle {{\underset {x=2}{\operatorname {limit} }}\;(x+2)=4}

;

and

therefore

\scriptstyle {{\underset {x=2}{\operatorname {limit} }}\;{\frac {x^{2}-4}{x-2}}=4}

.

therefore

Although the function is not defined for $\scriptstyle {x=2}$ , if we arbitrarily assign it the value $\scriptstyle {4}$ for $\scriptstyle {x=2}$ , it then becomes continuous for this value.

A function $\scriptstyle {f(x)}$ is said to be continuous in an interval when it is continuous for all values of $\scriptstyle {x}$ in this interval.^[1]

↑ In this book we shall deal only with functions which are in general continuous, that is, continuous for all values of $\scriptstyle {x}$ , with the possible exception of certain isolated values, our results in general being understood as valid only for such values of $\scriptstyle {x}$ for which the function in question is actually continuous. Unless special attention is called thereto, we shall as a rule pay no attention to the possibilities of such exceptional values of $\scriptstyle {x}$ for which the function is discontinuous. The definition of a continuous function $\scriptstyle {f(x)}$ is sometimes roughly (but imperfectly) summed up in the statement that a small change in $\scriptstyle {x}$ shall produce a small change in $\scriptstyle {f(x)}$ . We shall not consider functions having an infinite number of oscillations in a limited region.

[1] In this book we shall deal only with functions which are in general continuous, that is, continuous for all values of $\scriptstyle {x}$ , with the possible exception of certain isolated values, our results in general being understood as valid only for such values of $\scriptstyle {x}$ for which the function in question is actually continuous. Unless special attention is called thereto, we shall as a rule pay no attention to the possibilities of such exceptional values of $\scriptstyle {x}$ for which the function is discontinuous. The definition of a continuous function $\scriptstyle {f(x)}$ is sometimes roughly (but imperfectly) summed up in the statement that a small change in $\scriptstyle {x}$ shall produce a small change in $\scriptstyle {f(x)}$ . We shall not consider functions having an infinite number of oscillations in a limited region.

[1]