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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\R^n$ be an [[open set|02BH3MCBQB]], and let $C^\\infty_c(\\Omega)$ be the set of smooth functions $\\varphi \\colon \\Omega \\to \\R$ whose support is a compact subset of $\\Omega$.\nLet $f \\colon \\Omega \\to \\R$ be locally [[integrable|X7GTF3D98F]], that is, integrable on every compact subset of $\\Omega$.\nIf\n$$\\int_\\Omega f(x) \\varphi(x) \\, dx = 0 \\quad \\text{for every } \\varphi \\in C^\\infty_c(\\Omega),$$\nthen $f = 0$ [[almost everywhere|RARTMA0ZEC]] on $\\Omega$.\nIn particular, if $f$ is continuous, then $f(x) = 0$ for every $x \\in \\Omega$.\n\n## Remarks\n\nFor continuous $f$ the proof is short.\nIf $f(x_0) \u003e 0$, then $f \u003e 0$ on a ball $B$ around $x_0$; a [[bump function|HS65DW0HXH]] $\\varphi \\ge 0$ supported in $B$ with $\\varphi(x_0) \u003e 0$ gives $\\int_\\Omega f \\varphi \\, dx \u003e 0$, a contradiction.\nFor locally integrable $f$, one convolves $f$ with a smooth mollifier: the hypothesis makes every mollified function vanish, and these converge to $f$ in $L^1$ on compact sets.\n\nThe lemma is the step that turns the vanishing of the [[first variation|5B8NFYB2XQ]] into the [[Euler–Lagrange equation|PS0VV149ET]].\nAfter an integration by parts, the first variation has the form $\\int_\\Omega E(x) \\varphi(x) \\, dx$, where $E$ is the Euler–Lagrange expression; the lemma gives $E = 0$.\n\nThe class of test functions must be large enough.\nIf only $\\varphi$ with $\\int_\\Omega \\varphi \\, dx = 0$ were allowed, every constant $f$ would satisfy the hypothesis; this case is the [[du Bois-Reymond lemma|AATQKCWAAC]].\n\n## Explanations\n\n### L1 human\n\nIf a function gives zero when it is averaged against every smooth bump, then the function is zero.\nA bump placed where the function is positive would otherwise give a positive answer.\nIt is the reason why \"zero for all variations\" can be turned into an equation that holds at every point.\n\n### L2 human\n\nIn a derivation of the Euler–Lagrange equation one first finds $\\int_a^b \\big(L_z - \\frac{d}{dx} L_p\\big)\\eta \\, dx = 0$ for all $\\eta$ that vanish at the ends.\nThe lemma then gives the pointwise equation $L_z - \\frac{d}{dx} L_p = 0$.\nIn distribution theory the lemma says that a locally integrable function is determined by the distribution it defines.\n",
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