MathDB

Let

f,g:\mathbb{R}\to\mathbb{R}

be two nondecreasing functions.
[*]Show that for any

a\in\mathbb{R},

b\in[f(a-0),f(a+0)]

and

x\in\mathbb{R},

the following inequality holds

\int_a^xf(t) \ dt\geq b(x-a).

[*]Given that

[f(a-0),f(a+0)]\cap[g(a-0),g(a+0)]\neq\emptyset

for any

a\in\mathbb{R},

prove that for any real numbers

a<b

\int_a^b f(t) \ dt=\int_a^b g(t) \ dt.

Note:

h(a-0)

and

h(a+0)

denote the limits to the left and to the right respectively of a function

h

at point

a\in\mathbb{R}.

Problem Statement