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Differentiable function / showing a set is a neighbourhood of a point

  1. Jun 1, 2009 #1
    1. The problem statement, all variables and given/known data

    Suppose f:Rn ----> R is differentiable at the origin, (but not necessarily elsewhere), that f(0)=0, and that there is a constant c such that the norm of the gradient of f at zero is less than c. (||˅f(0)||<c )
    Show that the set U = {x e Rn : ||f(x)||< c||x|| } is a neighbourhood of the origin.

    Hint: Set epsilon to ε = c - ||˅f(0)||
    (thats c minus the norm of the gradient of f at zero...im not sure how to do a full upside down delta for the norm lol)



    3. The attempt at a solution
    First of all...i've no idea why I need an epsilon?? Is this a continuity proof cause it doesn't look like it...I really have no idea where to start
    the function being differentiable at the origin means that
    lim as h-->0 of [ f(0+h) - f(0) - ˅f(0)· h ] / |h| = 0

    so f(h) = f(0) + ˅f(0) = ˅f(0)

    and we have f(h) = ˅f(0)
    um...no i really have no idea what to do with this question :(
    help!
     
  2. jcsd
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