PomeRemark 5.15.5.3833CFAB6397 In the second case, we can always arrange that
\[ \varphi |_{(E\setminus \{ x\} )\cap U}=0. \]
In fact, we may assume that $\varphi \leq 0$ and $C<0$ is such that
\[ \varlimsup _{(U\cap E)\setminus \{ x\} \ni y\to x} \varphi (y)<C<\varphi (x). \]
We let
\[ \psi =(-C)^{-1}(\varphi \lor C)+1. \]
Then $\psi $ satisfies our requirements for a smaller $U$.
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