Vector of reduction

Vector of reduction

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ be a locally Lipschitz
continuous and $\bar{x}\in\mathbb{R}^2$. Put $$
S(x):=\left\{x^*\in\mathbb{R}^2: \liminf_{u\rightarrow
x}\frac{f(u)-f(x)-\langle x^*,u-x \rangle}{\|u-x\|}\geq 0\right\} $$ 1.
Suppose that $S(\bar{x})\ne\emptyset$ and $0\notin S(\bar{x})$. I would
like to know whether we can find a vector $d\in \mathbb{R}^2$ such that $$
f(\bar{x}+td)<f(\bar{x}) \quad {\rm for\; all\; t>0\; sufficiently\;
small.} $$ 2. Let $$ P(\bar{x}):=\limsup_{x\rightarrow
\bar{x}}S(x):=\left\{x^*\in \mathbb{R}^2: \exists\; x_k\rightarrow
\bar{x}, \; x^*_k\rightarrow x^*, \; x^*_k\in S(x_k), \;
k=1,2,\ldots\right\}. $$ Suppose that $P(\bar{x})\ne \emptyset$ and
$0\notin P(\bar{x})$. I would like to know whether we can find a vector
$d\in \mathbb{R}^2$ such that $$ f(\bar{x}+td)<f(\bar{x}) \quad {\rm for\;
all\; t>0\; sufficiently\; small.} $$ Note: If $f$ is differentiable at
$\bar{x}$, we can choose $d=\nabla f(\bar{x})$.
I would like to thank to all comments and construction.