Finding a direction with smaller values of a Lipschitz function 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 $d$ such that $\langle d,\nabla f(\bar{x})\rangle<0$.

The answer to the first question is no:
$$
f(x,y) = \cases{ -x & if $x \le 0$ \\ x \sin\log x & if $x > 0$ }
$$
has the unique subgradient $(-1, 0)$ where $x = 0$, but there are nonnegative values in every direction. (It looks very much like Robert Israels example, but
this one is Lipschitz)
The answer to the second question is also no:
$$
g(\theta, r) = \cases { 0 & if $r = 0$ \\ r \sin(\theta + \log r) & otherwise}
$$
is the expression in polar coordinates of a function with a gradient that
covers the same annulus in every punctured neighbourhood of 0 and does not
have a subgradient at 0. Clearly it oscillates about 0 in every direction.
I have not been able to find a common counterexample for both. It might be
that in the second case it would be enough to add the requirement that
$S(\bar x) \ne \emptyset$.
