Euler:1707-1783

One could claim that mathematical analysis began with
Euler.

In 1748 in *Introductio in analysin
infinitorum* Euler made ideas of __Johann Bernoulli__ more precise
in defining a function, and he stated that mathematical analysis was the
study of functions. This work bases the calculus on the theory elementary
functions rather than on geometric curves, as had been done previously.

Functions defined by formulae.

Conventions for notation of constants (numbers):
e and p (pi).

Bolzano 1781- 1848

(1817 paper) In this work ...
Bolzano ... did not wish only to purge the
concepts of limit,

convergence, and derivative
of
geometrical components and replace them by

purely
arithmetical concepts. He was aware of a *deeper
problem: the need to refine**
and enrich the concept of number itself.*

The paper gives a** proof of the
intermediate value theorem** with Bolzano's new approach and in
the work he **defined what is now called a Cauchy
sequence**. The concept appears in Cauchy's work four years later
but it is unlikely that Cauchy had read Bolzano's work.

**A bounded sequence has a convergent
subsequence.**

Cauchy:1789-1857

In 1817 when __Biot__
left Paris for an expedition to the Shetland Islands in Scotland Cauchy
filled his post at the Collège de France. There he lectured on methods
of integration which he had discovered, but not published, earlier.

Cauchy was the **first
to make a rigorous study of the conditions for convergence of infinite
series in addition to his rigorous definition of an integral.**His
text *Cours d'analyse* in 1821 was designed for students at Ecole
Polytechnique and was concerned with developing the basic theorems of the
calculus as rigorously as possible.

Conventions on limits, notation for derivatives, and sequence convergence.

Dirichlet (1805-1859)

Definition of function (1837)

*If a variable y is so related to a variable x that*
*whenever a numerical value is assigned to x,*
*there is a rule according to which a unique value
of y is determined,*
*then y is said to be a function of the independent
variable x.*

Dirichlet's function.

A.Problems with what is a function, law of excluded middle, what is continuous, what is integrable.f(x)=1 when x is rational,f(x)= 0 when x is not rational.fis not continuous at any x.

B. f(x)=1/q when x=p/q with gcd(p,q)=1, f(x)= 0 when x is not rational.f is not continuous at x rational, f is continuous at x not rational.