On fiber diameters of continuous maps

We present a surprisingly short proof that for any continuous map $f : $\mathbb{R}$^n \rightarrow $\mathbb{R}$^m$,if$n>m$, then there exists no bound on the diameter of fibers of $f$. Moreover, we show that when $m=1$, the union of small fibers of $f$ is bounded; when $m>1$, the union of small fibers need not be bounded. Applications to data analysis are considered.