Compact support vs. vanishing at infinity? Consider the two sets
$$ C_0 = \{ f: \mathbb R \to \mathbb C \mid f \text{ is continuous and } \lim_{|x|\to \infty} f(x) = 0\}$$
$$ C_c = \{ f: \mathbb R \to \mathbb C \mid f \text{ is continuous and }  \operatorname{supp}{(f)} \text{ is bounded}\}$$

Aren't these two sets the same? What am I missing?


The first one contains functions which are nonzero over the whole real line. The second one does not.
