DISCONTINUOUS FUNCTIONS. 237 



mathematicians discontinuous functions, and the 

 points C, C &c. are called breaks. The object 

 of the following researches is to express, by means 

 of one equation, the relation of the system of lines 

 generated by these discontinuous functions. 



(3). In the above exposition of discontinuity, 

 the independent variable is supposed to be con- 

 tinuous, or to admit of every magnitude between 

 the limits .v zz o and ,v — a. If the independent 

 variable be allowed to take only particular values 

 between the limits x — o and <.v — a, &c. &c. another 

 species of discontinuity will be brought into ope- 

 ration, a particular case of which will lead us to 

 the calculus of finite differences. 



(4). The following analytical explanation of 

 these different variations will tend to place the 

 subject in a clearer point of view. 



Let 1/ — f(a:) ; If we suppose cV to take an 

 increment A, and the function to remain constant, 

 we shall have, by putting ?/' to represent the cor- 

 responding increment of the function. 



f ' ( v) f " (x) 



If' = i{x + h) = f(.T) +~^h +~y;^-K' + &c. &c. 



