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Let $X$ be some subset of euclidean space and for the bounded function $f : X \to \mathbb{R}$ and let $f(x) = \sup f(X)$. Is the limit

$$\lim_{t \to 0} \frac{f(x+ta) - f(x)}{t}$$

if it exists guaranteed to be nonpositive?

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Hint: Suppose $f$ is constant. –  Ross Millikan Oct 16 '12 at 4:27
    
I have edited my question –  Kav Oct 16 '12 at 4:37
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1 Answer

No. Let $X=[0,1]$ and $f(x)=x$. Then $f(1)=\sup f(X)$, and

$$ \lim_{t\to 0}\frac{f(x+ta)-f(x)}t=\frac{x+ta-x}t=a\;, $$

which can be positive, zero or negative depending on $a$.

By the way, I think one question mark would have been enough.

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