nLab
bump function

Idea

A bump function is a smooth function with compact support, especially one that is not zero on a space that it not compact.

One reason the category SmthMfd of smooth manifolds and smooth functions is important is that a good supply of bump functions exists; this fact accounts for the great flexibility of smooth objects, in stark contrast to analytic geometry or algebraic varieties (since non-zero analytic or algebraic functions with compact support exist only on compact spaces). For instance, bump functions can be used to construct partitions of unity, which can in turn be used to smoothly patch together local structures into a global structure without obstruction, as for example Riemannian metrics.

Constructions

Define a function $\varphi (x)$ on the standard open unit ball of the cartesian space ${ℝ}^n$ by

$$\varphi (x)=\exp \left(\frac{1}{\parallel x{\parallel }^2-1}\right)$$\phi(x) = \exp\left( \frac1{{\|x\|^2} - 1} \right)

so that $\varphi (x)$ and all of its higher derivatives vanish rapidly as $x$ approaches the boundary. This gives a smooth function compactly supported on the unit ball? centered at the origin.

For $\epsilon >0$, define ${\varphi }_{\epsilon }(x)≔\varphi (x/\epsilon )$, and define

$${\psi }_{\epsilon }=\frac{{\varphi }_{\epsilon }}{{\parallel {\varphi }_{\epsilon }\parallel }_1}$$\psi_\varepsilon = \frac{\phi_\varepsilon}{{\|\phi_\varepsilon\|}_1}

so that the family $({\psi }_{\epsilon }{)}_{\epsilon }$ is a smooth approximation to the identity? (of convolution, the Dirac functional), compactly supported on the closed ball of radius $\epsilon$ and having an $L^1$-norm equal to $1$. Then, it is standard that for any pair $K\subset V$ with $K$ compact and $V$ open in a cartesian space ${ℝ}^n$, we can choose an open $U$ containing $K$ and with compact closure contained in $V$, and then taking the convolution product

$${\psi }_{\epsilon }*{\chi }_U$$\psi_\varepsilon \ast \chi_U

of ${\psi }_{\epsilon }$ with the characteristic function ${\chi }_U$, for $\epsilon$ sufficiently small, we obtain a smooth function equal to $1$ on $K$ and equal to $0$ outside $V$.

By performing the above construction in charts, we obtain, in any smooth manifold, a smooth function equal to $1$ on any compact subspace $K$ and equal to $0$ outside any neighbourhood $V$ of $K$. (This is a smooth regularity property.)

References

• wikipedia, bump function