Fundamental lemma of the calculus of variations

The fundamental lemma states that a continuous function whose integral against every admissible test function vanishes must itself vanish everywhere. It turns an integral identity into a pointwise conclusion.

Connect

Test function: A smooth function, often with compact support, used to probe a function or distribution through integration. Test functions localize the integral so a nonzero value cannot be hidden by cancellation.
Calculus of variations: The study of extrema and stationary values of functionals, often integrals depending on unknown functions. The fundamental lemma supplies the pointwise inference used throughout variational proofs.
Continuous function: A function whose values change arbitrarily little when its input changes sufficiently little. Continuity upgrades nonvanishing at one point to a neighborhood where a localized test detects it.
Du Bois-Reymond lemma: A lemma in the calculus of variations deriving pointwise conditions from integral identities, with versions allowing weaker regularity. Its formulations extend the fundamental lemma's role to settings where continuity is not assumed.
Bump function: A smooth function with compact support, commonly chosen to be nonnegative and positive near a selected point. A bump concentrated where the function is nonzero supplies the contradiction in a standard proof.
Euler–Lagrange equation: A differential equation satisfied by stationary functions of a variational integral under suitable regularity conditions. The lemma turns the first variation's vanishing for all perturbations into this equation.
Locally integrable function: A measurable function whose absolute value has finite integral on every compact subset of its domain. For merely integrable functions, vanishing against tests generally implies zero only almost everywhere.
Fundamental theorem of calculus: A theorem relating integration of a function to differentiation of its accumulation function. Both connect integral information to pointwise behavior, but by different hypotheses and mechanisms.
Compact support: A function has compact support when it vanishes outside some compact subset of its domain. Compact support lets the test function isolate an interior region without boundary contributions.
First variation: The first-order change in a functional caused by an infinitesimal perturbation of its argument. After integration by parts, a vanishing first variation leaves an integral to which the lemma applies.