I'm trying to understand this whole concept of uniform continuity. I've read the wikipedia article, I've gone through the section about it in my textbook, I've copied down the definition a few times, and i've gone through this pdf: http://www.math.wisc.edu/~robbin/521dir/cont.pdf
Technically speaking, I think I know what to do:
When asked to prove that:
$f(x) = x^2$ is uniformly continuous on $[0,3]$
I went through the steps of the definition and chose my $\delta$ to be $\epsilon/6$ (which is positive) and said:
Ok, assuming I choose an $x$ and $x_0$ from $[0,3]$ so that $|x_0 - x|<\delta$ then I must show that $|f(x_0) - f(x)|<\epsilon$ for every $\epsilon>0$.
$|x_0^2 - x^2| = (x_0 + x)|x_0-x| < \delta* (x_0+x) = \epsilon /6 *(x_0+x)< \epsilon /6 * (3 + 3) = \epsilon $
this part I don't really understand...am I just choosing the largest possible values for $x$ and $x_0$ on the interval? I'm also not sure what's the point of this whole thing, and it took me a bit of playing around with different values for $\delta$ until I got one that works and then realized there's a systematic way to calculate $\delta$ and then I wondered then what's the point of this whole proof delta/epsilon structure?
I've also read some of the previous questions and answers on uniform continuity and some of the explanations went over my head. Is there a simpler way to understand this aside from just memorizing what steps to go through when I see a problem like this?
I understand that uniform continuity is a "stricter" form of continuity than the regular one, since it describes the attribute of continuity for an entire interval and not merely one point.