How can I find a subset of a set with "half the size" of the original?
I am trying to solve this problem and I came across this post. The solution using the intermediate value theorem is neat but how do we know f define in such a way is continuous?
Thank you.
Since $f(x)=m(E\cap[-x,x])$, it follows that $$|f(x)-f(y)|=\left|\int_{\mathbf R}\mathbf 1_E(\mathbf 1_{[-x,x]}-\mathbf 1_{[-y,y]} )\mathrm dm\right| \leqslant \int_{\mathbf R}\left|\mathbf 1_{[-x,x]}-\mathbf 1_{[-y,y]} \right|\mathrm dm.$$ If $x\lt y$, then $\left|\mathbf 1_{[-x,x]}-\mathbf 1_{[-y,y]} \right|\leqslant\mathbf 1_{[x,y]} +\mathbf 1_{[-y,-x]}$, hence $$|f(x)-f(y)|\leqslant 2|x-y|.$$
