I'm having some problems understanding the following paragraph, which I read in a analysis script (hopefully I haven't made any translation errors):
"A map $f:U \rightarrow Y$, where $U$ is open and $X,Y$ are Banach spaces, is continuous at $x' \in U$ if $$f(x')=\lim_{x\rightarrow x'} f(x)=\lim_{h\rightarrow 0} f(x'+h),$$ where $h=x-x'$. We can decompose $h$ in a "polar" fashion in $h=ts$, where $\left\Vert h \right\Vert \geq 0$ and $s=\frac{1}{\left\Vert h \right\Vert } h$. Then $f(x')=\lim\limits_{h\rightarrow 0} f(x'+h)$ iff $f(x'+ts)\rightarrow f(x')$ for $t \rightarrow 0^+$ uniformly with respect to $\left\Vert s \right \Vert = 1$. No matter from which direction $s$ with $\left\Vert s \right\Vert=1$ we approach $x'$, the value of the function has to converge to $f(x')$ with a to all $s$ common "minimal speed" ".
What I don't understand is this:
1) What does in means to decompose anything in a "polar" fashion ?
2) I thought only sequences of function can converge uniformly...and what does it mean, if something converges uniformly with respect to another thing ?
3) What does the author mean with "common "minimal speed""
Thanks in advance.