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The noncontact set is where the unknown function is strictly above its obstacle. For continuous functions it is open. Every smooth variation with compact support in this set remains feasible with either sign when its amplitude is small enough. The first variation therefore vanishes there, yielding the unconstrained Euler-Lagrange equation. On the complementary contact set of an obstacle problem, only the appropriate one-sided variational inequality is available globally.

The noncontact set $D=\{x\in (0,l):u(x)>g(x)\}$ is open. For $\phi \in C_0^{\infty }(D)$, the positive gap above the obstacle has a positive minimum on the support of $\phi$. Thus both $u+t\phi$ and $u-t\phi$ are admissible for sufficiently small $t>0$. Applying the variational inequality to both signs yieldsHence the Euler-Lagrange equation on the noncontact set isIt holds classically under the assumed smoothness. On the contact set the obstacle prevents the negative variations used to obtain this equality, so a variational inequality remains the appropriate global description.