Note on the Intermediate Value Theorem (TVI)

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.