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Math Help - Directional Derivative implying Continuity

  1. #1
    Junior Member
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    Directional Derivative implying Continuity

    I'm having problems with how to prove this statement:

    Suppose that a function g:R^n-->R has the property that Dug(0)=0, where Dug(0) is the directional derivative of g in the direction of unit vector u at the point (0,...,0).
    Does it follow that g is continuous at 0? Prove or give an example showing that it is false.

    Thanks a lot!
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  2. #2
    Junior Member nimon's Avatar
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    Edinburgh, UK
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    Hi AKTilted,

    I have assumed in your question that \mathbf{u} can be any unit vector. If this isn't true then I don't think it can be shown.

    If this is true for any unit vector \mathbf{u}, then it is true for each \mathbf{u}_{i} = (0,\ldots,1,\ldots,0), where 1 is in the i^{\text{th}} position. Then we have that
    D_{u}g(0)=\nabla g(0) \cdot \mathbf{u}_{i} = \frac{\partial g}{\partial x_{i}}(0) = 0.
    Now since g does not change in any direction from 0, then it must be a constant function near 0, which is continuous.
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