Tangent cone of graph and epigraph sets.

Question

Let us first recall the definition of tangent cone $\; T(\bar x; \Omega)$ of a subset $\Omega$ at $\bar x \in \Omega$, where $X$ is a Banach space:

$$T(\bar x; \Omega)=\{v\in X:\; \; \exists \{x_k\}\subset \Omega, x_k\to \bar x, \exists \{t_k\}\subset (0; +\infty) , t_k\to 0 \; \mbox{ such that } \frac{x_k-\bar x}{t_k}\to v\}.$$

Consider the fucntion $f$ below

$$f(x)=x^2\sin(1/x) \; \mbox{ if } \; x\neq 0, \ \ \ \mbox{and} \ \ \ f(0)=0.$$

Determine the tangent cone of the following subsets of $X=\mathbb R^2$ at $\bar x=(0, 0)$:

(1) $\Omega_1=\mbox{epi}\; f:=\{(x; \gamma) \in \mathbb R\times \mathbb R: \; f(x)\leq \gamma\}.$

(2) $\Omega_2=\mbox{graph}\; f:=\{(x; f(x)):\;\; x \in \mathbb R\}$.

Here is my attempts: I consider the equation $x^2\sin(1/x)=ax$, $-1\leq a\leq 1$, $a\neq 0$. This implies, $x=0$ or $x\sin(1/x)=a$ but $\lim \limits_{x\to 0} x\sin(1/x)=0$ hence each line $y=ax$ in a sufficient small neighborhood of $0$ has only $x=0$ in commond with $\mbox{graph}\; f$ hence: $\; T(\bar x; \Omega_2)=\mathbb R\times \{0\}.$

I am not sure about $T(\bar x; \Omega_1)=\{(x_1, x_2):\;\; x_2\geq -|x_1|\}\; $? Please help me to solve this problem. I would highly appreciate your answer.

Answer 1

The answer $ T(\bar x; \Omega_2)=\mathbb R\times \{0\}$ is correct. The other one should be $$T(\bar x; \Omega_1)=\{(x_1, x_2):\;\; x_2\geq0\}$$

Let's observe two important (but easy to prove) properties of a tangent cone:

1. Locality: if $U$ is a neighborhood of $\bar x$, then $T(\bar x;\Omega\cap U)=T(\bar x;\Omega )$.
2. Monotonicity: if $\Omega\subseteq \widetilde\Omega$, then $T(\bar x;\Omega )\subset T(\bar x;\widetilde \Omega )$.

Next, observe that for every $\epsilon>0$, there is a neighborhood of $0$ in which $|f(x)|\le \epsilon |x|$. Therefore:

• the tangent cone of the graph of $f$ is contained in $\{(x_1,x_2):|x_2|\le \epsilon |x_1|\}$
• the tangent cone of the epigraph of $f$ is contained in $\{(x_1,x_2): x_2\ge -\epsilon |x_1|\}$

Since $\epsilon $ was arbitrarily small, the above holds also with $\epsilon$ replace by $0$.

To demonstrate the reverse inclusions, consider intersection of the half-lines contained in the purported cone with the relevant set. These intersections contain points arbitrarily close to the origin.