242, Outo State ACADEMY OF SCIENCE 
But if biologists did but know it, the characteristics, 
peculiarities 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 eo 
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 time 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,’ 1n which 
are also contained very interesting general conisderations on 
functions. 
“Tt 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.”’ 
