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Math Help - Derivative 0 => constant function

  1. #1
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    Derivative 0 => constant function

    Suppose that M, N are smooth manifolds and that M is connected. Let f: M \rightarrow N be differentiable so that d_x f = 0 for all x \in M.

    Show that f must be constant.


    Anyone got any hint? Why do we need M to be connected?
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  2. #2
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    If we don't require M to be connected, then we can let M=[0,1] \cup [2,3] ~, ~ N = [0,1] and f(x) = \left\{<br />
     \begin{array}{lr}<br />
       1 &  x \in [0,1]\\<br />
       0 &  x \in [2,3]<br />
     \end{array}<br />
   \right.
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  3. #3
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    Ok, I see. Now how to start proving this?
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  4. #4
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    Use proof by contradiction. Suppose that f(x)\ne f(y) and show that this leads to a non-zero derivative.
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