Du Bois-Reymond lemma

If $N$ is a continuous function on the segment $[x_1,x_2]$ and if for all continuously-differentiable functions $\eta$ which vanish at $x=x_1$, $x=x_2$ the relation

$$\int\limits_{x_1}^{x_2}\eta'(x)N(x)dx=0$$

is valid, then $N=\text{const}$ on $[x_1,x_2]$. Formulated by P. du Bois-Reymond [1].

The du Bois-Reymond lemma is employed in the calculus of variations to derive the Euler equation in its integral form. In this proof it is not necessary to assume that the extremum of the functional is attained on a twice-differentiable curve; the assumption of continuous differentiability is sufficient.

References

[1]	P. du Bois-Reymond, "Erläuterungen zu der Anfangsgründen der Variationsrechnung" Math. Ann. , 15 (1879) pp. 283–314
[2]	M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian)

References

[a1]	S. [S.V. Fomin] Fomine, "Commande optimale" , MIR (1982) (Translated from Russian)