Let the space $X=Y\cup\{a\}$, where $Y$ is a metriable subspace and $a \in \overline Y$. As we know, if $f$ is the coutinuous function from $Y^2$ to $I$, then $f$ can be continuously extendable over $X^2$. Since $Y$ is a metrizable space, there exist some continuous functions from $Y^2$ to $I$ satisfying $f(y\_1,y\_2)\neq 0$ whenever $y\_1 \neq y\_2$; $f(y\_1,y\_2)=0$ whenever $y\_1 = y\_2$. Therefore, the extensions $f^\sim (a,a)=0$. However, I don't know whether there exists one of those extensions satisfying that $f^\sim (a,y)\neq0$ and $f^\sim (y,a)\neq0$ for any $y \in Y$. Could someone help me?