The Converse of the Intermediate Value Theorem: from Conway to Cantor to Cosets and Beyond

The classical Intermediate Value Theorem (IVT) states that if f is a continuous real-valued function on an interval [a, b] ⊆ R and if y is a real number strictly between f(a) and f(b), then there exists a real number x ∈ (a, b) such that f(x) = y. The standard counterexample showing that the converse of the IVT is false is the function f defined on R by f(x) := sin( 1 x ) for x 6= 0 and f(0) := 0. However, this counterexample is a bit weak as f is discontinuous only at 0. In this note, we study a class of strong counterexamples to the converse of the IVT. In particular, we present several constructions of functions f : R → R such that f [I] = R for every nonempty open interval I of R (f [I] := {f(x) : x ∈ I}). Note that such an f clearly satisfies the conclusion of the IVT on every interval [a, b] (and then some), yet f is necessarily nowhere continuous! This leads us to a more general study of topological spaces X = (X, T ) with the property that there exists a function f : X → X such that f [O] = X for every nonvoid open set O ∈ T .