4. Let ƒ : R → R be integrable with respect to the Lebesgue measure. Show that the function g: [0, ∞) → R defined by g(t) = sup{ [ \ƒ(x + y) — f(x)|da : \y| ≤ t} R for t> 0 is continuous at t = 0.

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 12T
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4. Let f : R →R be integrable with respect to the Lebesgue measure. Show
that the function g : [0, 0) –→ R defined by
g(t) = sup{ / \f(x+ y) – f(x)|dx : |y| < t}
R
for t >0 is continuous at t = 0.
Transcribed Image Text:4. Let f : R →R be integrable with respect to the Lebesgue measure. Show that the function g : [0, 0) –→ R defined by g(t) = sup{ / \f(x+ y) – f(x)|dx : |y| < t} R for t >0 is continuous at t = 0.
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