TITLE:
Note on the Intermediate Value Theorem (TVI)
AUTHORS:
Mohamed Bekkali
KEYWORDS:
Real Analysis, Scattered-Chains, Countable Boolean Algebras, 0-Dimensional Space, Stone Space
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.15 No.9,
September
1,
2025
ABSTRACT: The Intermediate Value Theorem (for short, TVI) is a real analysis concept that one encounters in her/his College first calculus course. It is stated for a real continuous function,
f
, on a closed non empty interval
I⊂ℝ
. These two hypothesis, on the function and the interval, ensure that all values in between
min
I
(
f
)
and
max
I
(
f
)
are taken by
f
. In this note, we consider any non constant real function, say
g
, on an interval
I
of the real line
ℝ
so that
g
is continuous except on an order-scattered subset; under these conditions, the interior of
g(
I
)
is not empty.