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Question regarding continuity

  1. Mar 9, 2008 #1
    Hi, I have a quick question:

    When we talk about smooth functions (say a vector field on Rn), why must we usually restrict the domain to an open set in Rn?

    Thanks!
     
  2. jcsd
  3. Mar 9, 2008 #2

    mathwonk

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    ask yourself what it would mean to say the graph of a function defined on a one point set has a tangent line?

    similarly what would it mean to say the graph of a function on a curve in the plane has a tangent plane?

    smooth means there are tangent spaces to the graph, so the graph should be big enough to have a tangent plane.

    so the domain has to be kind of fat, i.e. open. in order to have the right dimension.
     
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