Lebl-Basic-Analysis-I · Problem 1.4.5

The key insight is that you don't need to map all of $[0, 1]$ differently—just "remove" the endpoints by sending them somewhere inside $(0, 1)$, then shift everything else to make room. Think of it as a continuous deformation: the endpoints are problematic, so identify where they should go, then use a smooth function (like a carefully chosen linear or rational map on subintervals) to reroute the interior points around them without gaps or overlaps.

The key insight is that you don't need to map all of $[0, 1]$ differently—just "remove" the endpoints by sending them somewhere inside $(0, 1)$, then shift everything else to make room. Think of it as a continuous deformation: the endpoints are problematic, so identify where they should go, then use a smooth function (like a carefully chosen linear or rational map on subintervals) to reroute the interior points around them without gaps or overlaps.