242 Ohio State Academy of Science 



But if biologists did but know it, the characteristics,, 

 pecuharities and methods of investigation for continuous func- 

 tions differ essentially from those for discontinuous functions. 



Our calculus assumed continuity in all its functions, and 

 also that differentiability was a necessary consequence of this 

 continuity. 



Lobachevski, the creator of the non-Euclidean geometry,, 

 emphasized the distinction between continuity and differentia- 

 bility, therein also being half a century in advance of his 

 contemporaries. 



The mathematicians of the eighteenth century did not touch 

 the quetsion of the relation between continuity and differentia- 

 bility, presuming silently that every continuous function is ea 

 ipso a function having a derivative. 



Ampere tried to prove this position, but his proof lacked 

 cogency. The question about the relation between continuity 

 and differentiability awoke general attention between 1870 and 

 1880, when Weierstrass gave an example of a function contin- 

 uous within a certain interval and at the same tirae having no 

 definite derivative within this interval (non-differentiable) . 



Meanwhile, Lobachevski already in the thirties showed 

 the necessity of distinguishing the "changing gradually" (in 

 our terminology: continuity) of a function and its " unbrokeness " 

 (now: differentiability). 



With especial precision did he formulate this difference in 

 his Russian Memoir of 1835: "A method for ascertaining the 

 convergence," etc. 



"A function changes gradually when its increment dimin- 

 ishes to zero together with the increment of the independent 

 variable. A function is unbroken if the ratio of these two incre- 

 ments, as they diminish, goes over insensibly into a new function, 

 which consequently will be a differential-coefficient. Integrals 

 must always be so divided into intervals that the elements under 

 each integral sign always change gradually and remain 

 unbroken." 



In more detail Lobachevski treated this question in - his 

 work, "On the convergence of trigonometric series," in which 

 are also contained very interesting general conisderations on 

 functions. 



" It seems," he writes, "that we cannot doubt the truth that 

 everything in the world can be represented by numbers, nor the 

 truth that every change and relation in it can be expressed by 

 analytic functions. At the same time a broad view of the theory 

 admits the existence of a dependence only in the sense that we 

 consider the numbers united with one another as if given 

 together." 



