Test function ($C_c^{\infty }$)

A test function is a smooth function with compact support. Weak and distributional equations are identities obtained by integrating against every test function.

A cutoff function is a smooth function that equals one on a chosen inner set and vanishes outside a slightly larger set. On concentric balls $B_r\subset B_R$, it can be chosen with $|D\eta |\le C/(R-r)$.

The space of test functions is $\mathcal{D}(U)=C_c^{\infty }(U)$. A sequence converges in $\mathcal{D}(U)$ when all its supports eventually lie in one compact subset of $U$ and every derivative converges uniformly there.

The space of test functions has the locally convex inductive-limit topology of the spaces of smooth functions supported in a fixed compact set $K$. Each ${\mathcal{D}}_K$ uses seminorms $p_m(\phi )={\max }_{|\alpha |\le m}\Vert {\partial }^{\alpha }\phi {\Vert }_{\infty }$. Sequential convergence means a common compact support and uniform convergence of every derivative. A linear functional on this space is a distribution exactly when its restriction to each ${\mathcal{D}}_K$ satisfies a finite order of a distribution estimate; the order and constant may depend on $K$.

Every distribution is a limit of test functions in weak convergence of distributions. With a mollifier ${\rho }_{\epsilon }$ and an expanding smooth cutoff function $\chi (x/j)$, the functions $\chi (x/j)(u\ast {\rho }_{1/j})(x)$ have compact support and converge to $u$. For each fixed test, the cutoff eventually equals one on its support, while the reflected mollifier approximates that test with all derivatives in a common compact set.

A function is locally integrable when its absolute value has finite integral over every compact subset of its domain. It defines a regular distribution by integration against test functions.