AS COMMONLY INVESTIGATED. 227 



With these remarks I may close this preUrainary outline of the subject, 

 which, I trust, will not be considered as without interest or unnecessarily pro- 

 lix by those who bear in mind the great importance of the development to 

 which it relates, and the erroneous views of it which analysts of the first rank 

 have taken. 



The same observations will apply to another subject, which has, indeed, been 

 only touched upon incidentally by any of the writers with whose works I am 

 acquainted ; I allude to the continuity of functions, and. the division of this con- 

 tinuity into classes. D'Alembert and Lagrange, in their discussions con- 

 cerning the vibrations of a musical cord, and, more recently, Fourier, in his 

 admirable theory of the propagation of heat, have given us some examples of 

 that species of arrangement which they term discontinuous functions; and 

 PoissoN, in resuming the inquiry treated by the two former, has shown that 

 the theories of the musical cord require that it should every where have that 

 species of continuity which gives a single tangent to each point. And I may 

 add that Mr. Murphy, in the first of his Memoirs on the Inverse Method of 

 Definite Integrals, has divided functions according to the number of their breaks, 

 giving analytical expressions for each class. But, as all these writers have re- 

 garded functions wherein the condition mentioned by Poisson is not fulfilled, as 

 discontinuous, I have to add here some remarks oil that subject. 



Section II. 



Continuity of Functions. 



The observations which I intend to make on this head will, perhaps, be con- 

 veniently introduced by quoting from a small pamphlet containing some ele- 

 mentary views of this kind, and which I had printed, but without publication, 

 during the course of last year. 



" Continuous functions we should suppose to be such as, in increasing from 

 one grandeur to another, passed by insensible gradations through every inter- 

 mediate value; but this must be understood with considerable limitation, since 

 the degree of continuity which serves, at present, as the type of that quality 

 in the theory of number is the continuity possessed by the ordinary functions 



