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?


Note that $C_c \subset C_0$, but $C_c \neq C_0$. For example, $f(x) = \dfrac{1}{x^2+1}$ belongs to $C_0$ but not $C_c$.
What you seem to be assuming is that $\lim_{|x|\to\infty}f(x) = 0$ implies that there is some $N > 0$ with $f(x) = 0$ for all $|x| > N$. This is not true, as the above example demonstrates. That is, a function can limit to zero at $\pm\infty$ without ever being zero.
