 Deterministic boundary in an upper-tail chance constraint

A deterministic demand has probability one of meeting or exceeding its mean. Thus replacing a positive-variance Gaussian chance constraint by its naive zero-variance limit with a non-strict capacity inequality loses the boundary condition. The example $X=0$, $C=0$, $\epsilon =e^{-1}$ satisfies $m+\varphi \sqrt{v}\le C$ for finite $\varphi$, but violates the chance requirement.