Who came up with the $\varepsilon$-$\delta$ definitions and the axioms in Real Analysis? I've seen a lot of definitions of notions like boundary points, accumulation points, continuity, etc, and axioms for the set of the real numbers. But I have a hard time accepting these as "true" definitions or acceptable axioms and because of this it's awfully hard to believe that I can "prove" anything from them. It feels like I can create a close approximation to things found in calculus, but it feels like I'm constructing a forgery rather than proving.
What I'm looking for is a way to discover these things on my own rather than have someone tell them to me. For instance, if I want to derive the area of a circle and I know the definition of $\pi$ and an integral, I can figure it out.

The rigorous formulations of the $\delta,\varepsilon$ definitions of the limit and continuity, as well as the $\varepsilon$ definition of the convergence of sequences, in their today's form, were developed by Karl Theodor Wilhelm Weierstrass (1815-1897). 
Weierstrass presented this rigorous formulation of Mathematical Analysis for the first time in the lectures of a course named Differential Rechnung during the academic year 1859-60 in the Königliche Gewerbeinstitut in Berlin (now Technical University of Berlin). (See also Wikipedia's article.)
Nevertheless, sufficiently rigorous (with today standards) definition of the limit was given by Bernard Bolzano in 1817. 
The first major step, however, towards a $\delta,\varepsilon$ definition appears in the work of Augustin Louis Cauchy Cours d'Analyse (1821), where he wrote: 
The function $f(x)$ is continuous with respect to $x$ between the given limits if, between these limits, an infinitely small increment in the variable always produces an infinitely small increment in the function itself. 
Although Cauchy never used $\delta,\varepsilon$ definitions, he occasionally used  $\delta,\varepsilon$ arguments in his proofs.
