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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1996 • Question 4

Question

Math 6210 • Real Analysis Fall 1996 • Question 4

Let \(f\in L^1(\mathbb R)\). Define a function \(\varphi \) on \(\mathbb R\) by the formula \(\varphi (x)=\int _{-\infty }^{x}f(t)\,dt\). Show that \(\varphi \) is a uniformly continuous function on \(\mathbb R\).

Raw LaTeX

Let \(f\in L^1(\mathbb R)\). Define a function \(\varphi\) on
\(\mathbb R\) by the formula
\(\varphi(x)=\int_{-\infty}^{x}f(t)\,dt\). Show that \(\varphi\) is a uniformly
continuous function on \(\mathbb R\).

Topic Tags

Lebesgue Integration Lp Spaces and Sequence Spaces