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

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

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 $$\scriptstyle{x=1}$$.

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.

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

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.