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.

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. f(x)=1 when x is rational, f(x)= 0 when x is not rational.
f is 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.