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notion of thevalueof thefunction diffring from thelimit 'by as little
as one wishes', a phrase which was taken up by Cauchy.
The origin of both Newton's and Leibniz' calculus was geometric
and it was only late in the eighteenth century that J. L. Lagrange (1797)
tried to make algebraic definitions of its fundamental notions. Having,
as he thought, proved that every function has a Taylor series he showed
that
f
(
a
h
)
f
(
a
)
hf
(
a
)
hr
(
h
) where
r
(
h
) tends to 0 as
h
tends to 0,
seemingly without reference to infinitesimals, and used this to prove (the
need for proof being itself a remarkable insight) that a differentiable
function with positive derivatives was increasing. It was a small step
from this to Cauchy's methods. It is to Lagrange that we owe the
notion of the derived function
f
(
x
) as against theratio of infinitsimals
dy
/
dx
, and also theword
derivative
.
In 1823, Cauchy offered an
neighbourhood description of
derivatives, and this largely resolved the problem of the definition of
limits and therefore made it possible to construct rigorous proofs using
limits. Cauchy used infinitesimals, but he defined them as variables
which were tending to zero, not as indivisibles as they had been
considered in the seventeenth century. However, continuous functions
were the context of Cauchy's work on differentiation, so that the
proposition that a differentiable function was necessarily continuous
was for Cauchy a matter of definition, not a theorem. Indeed some
followers of Cauchy believed that they had successfully proved that
continuous functions must be differentiable! Also Cauchy's functions
were continuous, or differentiable, on intervals, so that he did not
conceive of a function that was defined everywhere, but continuous at
only onepoint. Defining continuity and differentiability in terms of the
behaviour of functions at a single point followed the work of Riemann
(1854) and Weierstrass (1860). Likekwise until the time of Weierstrass,
with the awareness of the completeness of the real numbers and of the
possibility of continuous but non-differentiable functions, it was not
realised that the result of our qn 42 was distinct from that of qn 40 on
the invertibility of derivatives.
The advent of rigour did not come to the calculus all at once.
Distinguished mathematicians in the twentieth century have failed to
provide a rigorous treatment of the chain rule. (See Hardy, 9th edition,
ch. 6, ยง114.)
