# Strict inclusion for recession cone of closure of a convex set

Let $$C$$ be a nonempty closed convex subset of $$\mathbb{R}^n$$. The recession cone of $$C$$ is given by $$R_C=\left\lbrace d\in\mathbb{R}^n:x+td\in C, \forall t>0, \forall x\in C\right\rbrace.$$ It is shown that $$R_C$$ is a closed, convex cone in $$\mathbb{R}^n$$.

Let $$C$$ be a convex subset and $$\overline{C}$$ be the closure of $$C$$. We can show that $$R_C\subset R_{\overline{C}}\qquad \text{and}\qquad \overline{R_{C}}\subset R_{\overline{C}}.$$ I want to find examples where these inclusions are strict. I tried some $$C$$ but found no subset that satisfies. Please help me with some examples. Thank you very much.

P/s: in some books, I see that the closeness of $$C$$ in the recession cone definition is unnecessary.

The following set taken from here fulfills the inclusions strict: $$C = \left\lbrace(x,y) \mid 0 \leq x < 1, y \geq 1\right\rbrace \cup \left\lbrace(x,y) \mid 0 \leq x \leq 1, 0 \leq y \leq 1\right\rbrace.$$ Then $$R_C = \{0\}$$ but $$R_{\overline{C}}$$ is the cone generated by the direction $$(0,1)$$.