How do we gain a better intuition on the definition of uniform continuity and its advantages compared to usual continuity? So here I am studying uniform continuity. Besides its definition, it has been proved that uniformly continuous functions map pairs of equivalent sequences to pairs of equivalent sequences, Cauchy sequences onto Cauchy sequences and bounded set onto bounded sets. Moreover, if a continuous function is defined on a closed and bounded interval, it is also uniformly continuous.
However I would like to gain a better intuition on its definition and know the advantages of this concept compared to usual continuity.
Any contribution is appreciated.
 A: In the first round continuity is defined at individual points. A function $f: \>X\to Y$ is continuous at the point $p\in X$, if for each $\epsilon>0$ there is a $\delta>0$ such that
$$x\in U_\delta(p)\ \Rightarrow \ f(x)\in U_\epsilon\bigl(f(p)\bigr)\ .$$
This definition creates a certain dependence $\epsilon\rightsquigarrow\delta$ that describes how large distances of $x$ from $p$  guarantee a value error of $f(x)$ from $f(p)$ which is still $<\epsilon$.
It is then said that the function $f$ is continuous on the space $X$ if $f$is continuous at each point $p\in X$. This sounds innocouus. But it means that for such a spacewide continuity of $f$ we have to have "uncountably many" dependencies $\epsilon\rightsquigarrow\delta$ under control, one for each point $p\in X$.
Now in proofs handling about continuous functions, e.g., about the existence of the Riemann integral, we want just one such dependency, in order to simplify matters. That's where uniform continuity comes in. The function $f$ is uniformly continuous on $X$,  if for each $\epsilon>0$ there is a $\delta>0$ such that for all $p\in X$ we have
$$x\in U_\delta(p)\ \Rightarrow \ f(x)\in U_\epsilon\bigl(f(p)\bigr)\ .$$
(When you don't like neighborhoods you can write $|x-p|<\delta$ instead of $x\in U_\delta(p)$.)
