Lebl-Basic-Analysis-I · Problem 3.6.13

The key is to "lift" $f$ slightly while keeping it trapped between the boundary constraints. Since $f$ is increasing but not strictly, you need to add a small, carefully controlled perturbation that breaks ties without overshooting the upper bound of $(0,1)$. Think of it as threading a strictly increasing curve through the "slack space" above $f$ and below $1$—the $\epsilon$ tolerance on the endpoint gives you just enough room to make this work for any initial gap.

The key is to "lift" $f$ slightly while keeping it trapped between the boundary constraints. Since $f$ is increasing but not strictly, you need to add a small, carefully controlled perturbation that breaks ties without overshooting the upper bound of $(0,1)$. Think of it as threading a strictly increasing curve through the "slack space" above $f$ and below $1$—the $\epsilon$ tolerance on the endpoint gives you just enough room to make this work for any initial gap.