Let's define the following set of functions:
$$C_c(\mathbb{R}) = \{f \in C(\mathbb{R}) | f(t) = 0, \forall t \text{ s.t. } |t| \geq T, T \geq 0\} $$
Then, can I say that for a general $f \in C_c(\mathbb{R}) $, $f$ is defined in $[-T,T]$ ? Because actually a general function is defined in all $\mathbb{R}$, but it is $0 \text{ } \forall \text{ } t \text{ } \geq T$.
(I need to say that because I want to use the fact that $f$ is defined in a compact set).