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.
 A: Contrary to a common misconception, one will not find an epsilon, delta definition of continuity in Cauchy even if you look with a microscope.  On the other hand, you will find his definition of continuity in terms of infinitesimals: every infinitesimal increment $\alpha$ necessarily produces an infinitesimal change $f(x+\alpha)-f(x)$ in the function.  More specifically, the recent translation
Bradley, Robert E.; Sandifer, C. Edward Cauchy's Cours d'analyse. An annotated translation. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York, 2009
contains the following material on Cauchy's definition. Cauchy's Section 2.2 is entitled Continuity of functions. Cauchy writes: "If, beginning with a value of $x$ contained between these limits, we add to the variable $x$ an infinitely small increment $\alpha$, the function itself is incremented by the difference $f(x+\alpha)-f(x)$",
and states that "the function $f(x)$ is a continuous function of $x$ between the assigned limits if, for each value of $x$ between these limits, the numerical value of the difference $f(x+\alpha)-f(x)$ decreases indefinitely with the numerical value of $\alpha$." Cauchy goes on to provide an italicized definition of continuity in the following terms: 
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.
A: 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.
A: Well, originally, definition of differential and integral used infintesimal numbers, but they were unknown and seemed nonsense, so they came up with the $\varepsilon$-$\delta$ definitions of limit, continuity and others, to make it precise. (Later those 'infinitesimal numbers' were found, in the sense that the set of real numbers can be nicely extended with them, staying logically consequent.)
But, this was just a middle step in the story, at least for limit and continuity. E.g., in general topology, it turned out that the most effective definition of continuity of a function $f:A\to B$ is that

The preimage $f^{-1}(V)$ of any open subset $V$ of $B$ is an open subset of $A$. $\quad\quad \quad\quad(1)$

In any metric space $(X,d)$, (i.e. where $X$ is a set and $d$ stands for 'distance' defined for pair of elements of $X$) the open subsets are defined to be the unions of (arbitrarily many) open balls $B_x(r):=\{y\in X\mid d(x,y)<r\}$. (In particular, in $\Bbb R$ the distance is given by $d(x,y):=|x-y|$ and $B_x(r)$ is the open interval $(x-r,x+r)$.)
Try to prove that the definition $(1)$ coincides with the $\varepsilon$-$\delta$ definition for $\Bbb R\to\Bbb R$ functions. 
(Hints: taking preimage preserves arbitrary unions, so we can reduce to the case when $V$ itself is an open ball.)
