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The converse of the intermediate value theorem: from Conway to Cantor to cosets and beyond. (English) Zbl 1311.26002

Summary: The classical intermediate value theorem (IVT) states that if \(f\) is a continuous real-valued function on an interval \([a,b]\subseteq\mathbb{R}\) and if \(y\) is a real number strictly between \(f(a)\) and \(f(b)\), then there exists a real number \(x\in(a,b)\) such that \(f(x)=y\). The standard counterexample showing that the converse of the IVT is false is the function \(f\) defined on \(\mathbb{R}\) by \(f(x):=\sin(\frac{1}{x})\) for \(x\neq 0\) 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 \(f \colon \mathbb{R}\to\mathbb{R}\) such that \(f[I]=\mathbb{R}\) for every nonempty open interval \(I\) of \(\mathbb{R}\) (\(f[I]:=\{f(x):x\in I\}\)). 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 \(X=(X,\mathcal{T})\) with the property that there exists a function \(f \colon X\to X\) such that \(f[O]=X\) for every nonvoid open set \(O\in\mathcal{T}\).

MSC:

26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54C10 Special maps on topological spaces (open, closed, perfect, etc.)

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