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Off the top of my head, all of these problems can be alleviated by considering a smooth map [itex]h:M \times (0,1) \rightarrow N[/itex] such that for some [itex]x,y \in (0,1)[/itex] with [itex]x < y[/itex] it follows that [itex]h(s,x) = f(s)[/itex] and [itex]h(s,y) = g(s)[/itex]. This is a bit fussier, but it is immediately clear that [itex]M \times (0,1)[/itex] is a smooth manifold.