Basic Analysis I — Real Analysis · Problem 7.5.18

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · continuous-functions-metric-spaces

Problem

Prove a stronger version of Proposition 7.5.12: If

f: (a,b) \times (c,d) \to \mathbb{R}

is a bounded continuous function, then

g: (c,d) \to \mathbb{R}

defined by

g(y) \coloneqq \int_a^b f(x,y)\,dx \quad \text{is continuous.}

Hint: First integrate over

[a+1/n,\, b-1/n]