The definitions of the two are so alike, that it confuses me.
Cauchy's General Principle:
The necessary and sufficient condition that a function $f(x)$ may tend to a definite limit, say $l$, as $x \to a$, is that:
If $\forall \epsilon>0$, $\exists$ a $ \ \delta>0$ such that $|f(x_1)-f(x_2)| < \epsilon$ whenever $0<|x_1-a|<\delta, \ 0<|x_2-a|<\delta $
Uniform Continuity:
A function $f$ defined on a domain $D$ is said to be uniformly continuous on the set $S$ $( S \subset D)$ if:
$\forall \ \epsilon >0$, $\exists $ a $\delta>0$, such that $|f(x_1)-f(x_2)|< \epsilon$ for any two points $x_1, x_2 \in S$ with $|x_1-x_2| < \delta$.
First of all, I understand that the former definition is a "Local Concept" i.e. it focuses around the particular point $a$ , while the other one is a "Global Concept", i.e. defined all over the set $S$. Another thing is, the sign "$0<$ " in the General Principle. It deletes the point "$a$" from the neighbourhood of both $x_1$ and $x_2$, while the other does not. But the similarity is, both definitions give independence to choose arbitrary pair of points within a certain neighbourhood.
Is anything wrong with my perception? Am I missing anything very obvious?
Please do provide further insight.
I get it that way:
simple convergence : given $a \in I$, $\forall \epsilon>0, \exists \delta$ such as $|x-a|<\delta \implies |f(x)-f(a)|<\epsilon$
therefore this definition is local, we defined $a$ before epsilon so every epsilon is a function of $a$, and every delta is a function of epsilon.
In your definition of uniform continuity, x1 and x2 are introduced after epsilon and delta, (not x dependents) which guarantees the fonction converges at every x at the same "speed".