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Thread: Proving a claim regarding differentiability

  1. #1
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    Proving a claim regarding differentiability

    Let F(x,y,z) be a function which is defined in the point M_0(x_0,y_0,z_0) and around it and the following conditions are satisfied:

    1. F(x_0,y_0,z_0)=0
    2. F has continuous partial derivatives in M_0 and around it
    3. F'_z(x_0,y_0,z_0)=0
    4. gradF at (x_0,y_0,z_0) != 0
    5. It is known that there is a function f(x,y) so that F(x,y,f(x,y)) =0 in M_0 and around it

    Prove that f(x,y) is not differentiable in (x_0, y_0)
    Last edited by GIPC; Jan 19th 2011 at 06:21 AM.
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  2. #2
    MHF Contributor

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    So, it is the "f" in (5) that you want to prove differentiable and not F?

    Do you recall the definition of "differentiable" for a function of two variables?

    f(x, y) is differentiable at (x_0, y_0) if and only if there exist a linear function L(x,y)= ax+ by and a function \epsilon(x,y) such that
    1) f(x, y)= F(x_0, y_0)+ a(x- x_0)+ b(y- y_0)+ \epsilon(x,y) and
    2) \lim_{(x,y)to (x_0,y_0)} \frac{\epsilon(x,y)}{\sqrt{(x-x_0)^2+ (y-y_0)^2}}= 0

    The demominator in (2) can be replaced with any reasonable measure of "distance" from (x_0, y_0)
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  3. #3
    Junior Member
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    I know the definition but I'm skeptical whether it helps here.

    I should probably use the chain rule somehow. Can't really point my finger how though.

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