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.