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?


Another example would be $f : x \mapsto e^{-|x|},$ $x \in \mathbb R.$ Then $f \in C_0 (\mathbb R)$ but $f \notin C_c (\mathbb R).$
