MathDB

An ineq. involving sum of def. int. as a characterization of locally cont. func.

Source: Romanian National Olympiad 2016, grade xii, p.3

August 25, 2019

functioncalculusintegrationinequalities

Problem Statement

Let be a real number

a,

and a nondecreasing function

f:\mathbb{R}\longrightarrow\mathbb{R} .

Prove that

f

is continuous in

a

if and only if there exists a sequence

\left( a_n \right)_{n\ge 1}

of real positive numbers such that

\int_a^{a+a_n} f(x)dx+\int_a^{a-a_n} f(x)dx\le\frac{a_n}{n} ,

for all natural numbers

n.

Dan Marinescu