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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 = 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 .

Abstract. We determine the relationship between the cardinality of a Noetherian integral domain and the cardinality of a residue field. One consequence of the main result is that it is provable in ZFC that there is a Noetherian domain of cardinality ℵ 1 with a finite residue field, but the statement "There is a Noetherian domain of cardinality ℵ 2 with a finite residue field" is equivalent to the negation of the Continuum Hypothesis.

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