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

.

For , . Moreover, if approaches the limit in any manner, the function approaches as a limit. Hence the function is continuous for .

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

Theorem. If is not defined for , and if

,

then will be continuous for , if is assumed as the value of for . Thus the function

is not defined for (since then there would be division by zero). But for every other value of ,

;

and
;
and
therefore
.
therefore

Although the function is not defined for , if we arbitrarily assign it the value for , it then becomes continuous for this value.

A function is said to be continuous in an interval when it is continuous for all values of in this interval.[1]

  1. In this book we shall deal only with functions which are in general continuous, that is, continuous for all values of , with the possible exception of certain isolated values, our results in general being understood as valid only for such values of 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 for which the function is discontinuous. The definition of a continuous function is sometimes roughly (but imperfectly) summed up in the statement that a small change in shall produce a small change in . We shall not consider functions having an infinite number of oscillations in a limited region.