A basic doubt on uniform continuity

Question

I can prove formally that the function $f(x)=\frac{1}{x^2}$ is not uniformly continuous in $(0,\frac{1}{1000})$, but uniformly continuous in $(\frac{1}{1000},1)$. But don't understand it intuitively. Is it because of the point 0 in the first case. Another point, it looks like uniform continuity does not imply uniform growth rate. In the second example (i.e $(\frac{1}{1000},1)$)near to left end point the function increases rapidly, but near to the right end point the increase is not that much. What is uniform then ?