Measure, Integration & Real Analysis · Problem 2B.23

Measures — Sheldon Axler

Mathematics — Measure Theory · monotone-functions

Problem

Suppose

f : \mathbf{R} \to \mathbf{R}

is a strictly increasing function. Prove that the inverse function

f^{-1} : f(\mathbf{R}) \to \mathbf{R}

is a continuous function. [Note that this exercise does not have as a hypothesis that

f

is continuous.]