Measure, Integration & Real Analysis · Problem 2E.9

Integration — Sheldon Axler

Mathematics — Measure Theory · continuity-on-union-of-closed-sets

Problem

Suppose

F_1, \ldots, F_n

are disjoint closed subsets of

\mathbf{R}

. Prove that if

g: F_1 \cup \cdots \cup F_n \to \mathbf{R}

is a function such that

g|_{F_k}

is a continuous function for each

k \in \{1, \ldots, n\}

, then

g

is a continuous function.