But why the problem says that "prove it" with such special $\varphi$?
2026-04-03 12:55:03.1775220903
I'm confused of the test problem of Weak Derivatives
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The $\varphi$ in question does not necessarily have compact support in $\mathbb{R}\times (0,\infty)$, but it vanishes only in $(x,0)$, which is a boundary point of the (open) domain of definition. So the definition does not apply.