The two-point duplication property of a function family relative to a target function

Let $X$ be a topological space, let $F\subseteq C(X,\mathbb{R})$ be a family of continuous real-valued functions (Continuity of a map of topological spaces at a point and globally, The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$), and let $f\in C(X,\mathbb{R})$. The family $F$ has the two-point duplication property relative to $f$ when for every $x,y\in X$, including $x=y$, there is $h\in F$ such that $h(x)=f(x)\text{and}h(y)=f(y)\mathrm{.}$

The equal-point clause matters on a one-point space. Erdman's distinct-point convention is equivalent to this one when $F$ contains every constant function, because the constant function with value $f(x)$ supplies the witness when $x=y$. Without constants, the distinct-point condition is vacuous on a singleton and does not imply approximation of an arbitrary target there.