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Homework Help: Question regarding Gateaux Derivative

  1. Apr 7, 2013 #1
    1. The problem statement, all variables and given/known data
    I am trying to solve the following problem:

    Let [itex]X[/itex] be space of continuous functions on [0,1] and let [itex]F:X\rightarrow\mathbb{R}[/itex] be defined by [itex]F(f)=\max\limits_{0\leq x\leq 1} f(x)[/itex] for any [itex]f\in X[/itex]. Show that the Gateaux Derivative does not exist if [itex]f[/itex] achieves a maximum at two different points [itex]x_1,x_2[/itex] in [0,1].

    2. Relevant equations
    The Gateaux Derivative for [itex]f,h\in X[/itex] is given by


    if the above limit exists for any increment [itex]h[/itex].

    3. The attempt at a solution
    Using the limit definition of the Gateaux Derivative, we see that


    This seems to work regardless of whether or not the function has a unique maximum, so that is the part I don't understand. Any help would be appreciated.
    Last edited: Apr 7, 2013
  2. jcsd
  3. Apr 7, 2013 #2


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    Science Advisor
    Homework Helper

    While I'm still not clear on the whole problem, I'll tell you one thing that's wrong. max(f(x)+th(x)) is not generally equal to max(f(x))+max(th(x)). Try it with a simple example like f(x)=x and h(x)=(-x).
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