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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2013 • Question 1

Question

Math 6210 • Real Analysis Spring 2013 • Question 1

Let \(f\in L^p(\mathbb R)\) (with respect to the Lebesgue measure), and \(1\le p\le \infty \), and let \[ F(x)=\int _0^x f(t)\,dt. \] Prove that \(F\) defines a continuous function on \(\mathbb R\).

Raw LaTeX

Let \(f\in L^p(\mathbb R)\) (with respect to the Lebesgue measure),
and \(1\le p\le\infty\), and let
\[
F(x)=\int_0^x f(t)\,dt.
\]
Prove that \(F\) defines a continuous function on \(\mathbb R\).

Topic Tags

Lebesgue Integration Lebesgue Measure and Translation Invariance Lp Spaces and Sequence Spaces Measure Theory