Question regarding Gateaux Derivative

[naericson]

Homework Statement

I am trying to solve the following problem:

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

Homework Equations

The Gateaux Derivative for $f,h\in X$ is given by

$\lim\limits_{t\to0}\frac{1}{t}\left(F(f+th)-F(f)\right)$

if the above limit exists for any increment $h$.

The Attempt at a Solution

Using the limit definition of the Gateaux Derivative, we see that

$\lim\limits_{t\to0}\frac{1}{t}\left(F(f+th)-F(f)\right)<br /> =\lim\limits_{t\to0}\frac{1}{t}\left(\max\limits_{x}(f+th)(x)-\max\limits_{x}f(x)\right)$
=$\lim\limits_{t\to0}\frac{1}{t}\left(\max\limits_{x}f(x)+\max_{x}th(x)-\max\limits_{x}f(x)\right)$
=$\lim\limits_{t\to0}t\frac{\max\limits_{x}h(x)}{t}$
=$\max\limits_{x}h(x)$.

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.

 

[Dick]

Homework Statement

I am trying to solve the following problem:

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

Homework Equations

The Gateaux Derivative for $f,h\in X$ is given by

$\lim\limits_{t\to0}\frac{1}{t}\left(F(f+th)-F(f)\right)$

if the above limit exists for any increment $h$.

The Attempt at a Solution

Using the limit definition of the Gateaux Derivative, we see that

$\lim\limits_{t\to0}\frac{1}{t}\left(F(f+th)-F(f)\right)<br /> =\lim\limits_{t\to0}\frac{1}{t}\left(\max\limits_{x}(f+th)(x)-\max\limits_{x}f(x)\right)$
=$\lim\limits_{t\to0}\frac{1}{t}\left(\max\limits_{x}f(x)+\max_{x}th(x)-\max\limits_{x}f(x)\right)$
=$\lim\limits_{t\to0}t\frac{\max\limits_{x}h(x)}{t}$
=$\max\limits_{x}h(x)$.

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.

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).