If I have a convex set $ \min f(x) $
$\sum_{i=1}^n x_{i} =1$ where $x\geq 0$.
Will an $\overline x$ (a local minimizer), if $x_{i} >0$ be
$ \dfrac{\partial{f(x)}}{\partial{x_{i}}} \geq \dfrac{\partial{f(x})}{\partial{x_{j}}} $ for all $j$
1)why would that be so ?
2) Does it matter if $f: \Bbb{R}^{n} \to \Bbb{R}$ is continuously differntiable ?
I assume you meant $$\frac{\partial f}{\partial x_i}(\bar{x}) \geq \frac{\partial f}{\partial x_j}(\bar{x})?$$
Notice that the constraints are symmetric in the unknowns. Without some more information about $f$, there is no reason an asymmetry must appear in the derivatives of $f$ at $\bar{x}$.