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We prove that every smooth closed manifold admits a smooth real-valuedfunction with only two critical values. We call a function of this type a\emph{Reeb function}. We prove that for a Reeb function we can prescribe theset of minima (or maxima), as soon as this set is a PL subcomplex of themanifold. In analogy with Reeb's Sphere Theorem, we use such functions to studythe topology of the underlying manifold. In dimension $3$, we give acharacterization of manifolds having a Heegaard splitting of genus $g$ in termsof the existence of certain Reeb functions. Similar results are proved indimension $n\geq 5$.

On smooth functions with two critical values