Let $M$ be a connected manifold with boundary and let $f: M \to \mathbb{R}$ be a Morse function (that does not necessarily send the boundary of $M$ to a point). Is it always possible to homotope $f$ through Morse functions so that all of the critical points are on the boundary of $M$?
My initial thought was that this can not be true since then the interior of $M$ admits a Morse function that does not have any critical points - but there is no problem with that as all open manifolds admit Morse functions without critical points.