Let $X$ be [0,1], function $\phi:X\times X\to\mathbb{R}^+$ is Lipschitz continuous on $X\times X$. Define $$ G = supp(\phi) = \overline{\{(x,y)|\phi(x,y)>0\}}. $$ and its section $$ s(x) = \{y\in X|(x,y)\in G\}. $$
Is it true that $s(x) = \overline{\{y\in X|\phi(x,y)>0\}}$?
Edited: Still is it true that $\overline{\{y\in X|\phi(x,y)>0\}}\subset s(x)$? If yes, what can we say about $s(x)\setminus \overline{\{y\in X|\phi(x,y)>0\}}$?
Edited2: Are we sure now that $\partial A'$ is of measure zero, where $$ \bigcup\limits_{x\in A}s(x) $$ for an arbitrary set $A$?