Support

The support of a function (Def. 5.1.1 ) is the smallest set outside which the function is equal to zero:

$$\text{supp} f= \bar{\{x\in R: f(x) \neq = 0\}}$$

If $$\text{supp}f$$ is bounded (i.e., contained in an interval in the form [a,b] for some $$a,b \in \mathbb{R}$$), then we say that $$f$$ has compact support.

Special vector spaces
There are two special vector spaces w.r.t. support.
 * $$C_c(\mathbb{R})$$, the space of all continuous functions with compact support: $$C_c(\mathbb{R}):=\left\{f:\mathbb{R}\to\mathbb{C} : f \text{ is continuous and has compact support}\right\}$$
 * $$C_0(\mathbb{R})$$, the space of all continuous functions that tend to zero as $$x\to\pm\infty$$: $$C_0(\mathbb{R}):=\left\{f:\mathbb{R}\to\mathbb{C} : f \text{ is continuous and } f(x)\to 0 \text{ as } x\to\pm\infty\right\}$$