differentiability). Sometimes the given functions are assumed to be piecewise continuous, in which case Riemann integration suffices, and the conclusions are stated everywhere except the finite set of discontinuity points. Higher derivatives If a tuple of continuous functions on an interval (a,b) satisfies the equality for all compactly supported smooth functions h on (a,b), then there exist continuously differentiable functions on (a,b) such that everywhere. This necessary condition is also sufficient, since the integrand becomes The case n = 1 is just the version for two given functions, since and thus, In contrast, the case n=2 does not lead to the relation since the function need not be differentiable twice. The sufficient condition is not necessary. Rather, the necessary and sufficient condition may be written as for n=2, for n=3, and so on; in general, the brackets cannot be opened because of non-differentiability. Vector-valued functions Generalization to vector-valued functions is straightforward; one applies the results for scalar functions to each coordinate separately, or treats the vector-valued case from the beginning. Multivariable functions If a continuous multivariable function f on an open set satisfies the equality for all compactly supported smooth functions h on Ω, then f is identically zero. Similarly to the basic version, one may consider a continuous function f on the closure of Ω, assuming that h vanishes on the boundary of Ω (rather than compactly supported). Here is a version for discontinuous multivariable functions. Let be an open set, and satisfy the equality for all compactly supported smooth functions h on Ω. Then f=0 (in L''2, that is, almost everywhere). Applications This lemma is used to prove that extrema of the functional are weak solutions (for an appropriate vector space ) of the Euler–Lagrange equation The Euler–Lagrange equation plays a prominent role in classical mechanics and differential geometry. Notes References (transl. from Russian). Classical mechanics Calculus of variations Smooth functions Lemmas in analysis