MathDB

Let

f,F:\mathbb{R}\longrightarrow\mathbb{R}

be two functions such that

f

is nondecreasing,

F

admits finite lateral derivates in every point of its domain,

\lim_{x\to y^-} f(x)\le\lim_{x\to y^-}\frac{F(x)-F\left( y \right)}{x-y} ,\lim_{x\to y^+} f(x)\ge\lim_{x\to y^+}\frac{F(x)-F\left( y \right)}{x-y} ,

for all real numbers

y,

and

F(0)=0.

Prove that

F(x)=\int_0^x f(t)dt,

for all real numbers

x.

Problem Statement