being assigned constants). For every value of
a definite value corresponds for
. A more complicated function would be given by
It was soon discovered that, if instead of annexing merely a large number of successive terms to the expression of our function we added an infinite number of terms selected in an appropriate manner, we often obtained functions presenting exceedingly simple properties; so simple, indeed, as to warrant our coining separate names for them. These various functions may be likened to different beings, possessing various peculiarities and tendencies. Certain mathematicians (Borel, for instance) have gone so far as to differentiate between wholesome functions and contaminated ones; and the appellation “pathology of functions” has been introduced into mathematics. As an example of the very simplest type of functions, we may mention those given by the unending series
where 3! stands for
