Every subfield of $\mathbb{R}$ contains $\mathbb{Q}$, hence is dense — so continuity forces the zero function

Let $X=\{f:\mathbb{R}\to\mathbb{R}:f\text{ is continuous such that }f(t)=0\text{ for all }t\text{ in a subfield of }\mathbb{R}\}$. Then X contains: