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?
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?