MathDB

Yet another convergence problem

Source: Romanian District Olympiad 2024 11.4

March 10, 2024

real analysislimit

Problem Statement

Consider the functions

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

such that

f{}

is continous. For any real numbers

a<b<c

there exists a sequence

(x_n)_{n\geqslant 1}

which converges to

b{}

and for which the limit of

g(x_n)

as

n{}

tends to infinity exists and satisfies

f(a)<\lim_{n\to\infty}g(x_n)<f(c).

[*]Give an example of a pair of such functions

f,g

for which

g{}

is discontinous at every point. [*]Prove that if

g{}

is monotonous, then

f=g.

Back to Problems View on AoPS