AKG
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I wasn't using the Rn -> Rm notation as a substitute for M -> N. I was using it because it was what I thought you meant by the "basic definition from simple calculus". I only used Rn -> Rm to talk about the simple calculus definition of continuity, never the manifold definition. And the notation Rn -> Rm DOES mean that every point in Rn gets mapped to a point in Rm, just as M -> N means that every point in M gets mapped toa point in N.