Definition of the notation $C_c^\infty (I)$?

Question

What is $C_c^\infty (I)$ where I is an interval in $\Bbb R$?

I think it can be:

1. The set of all infinitely many times differentiable functions $f:I\to \Bbb R$ where $\{ x\in I \mid f(x) \ne 0\}$ is compact.
2. The set of all infinitely many times differentiable functions $f:I\to \Bbb R$ where the closure of $\{ x\in I \mid f(x) \ne 0\}$ in I is compact.
3. The set of all infinitely many times differentiable functions $f:I\to \Bbb C$ where $\{ x\in I \mid f(x) \ne 0\}$ is compact.
4. The set of all infinitely many times differentiable functions $f:I\to \Bbb C$ where the closure of $\{ x\in I \mid f(x) \ne 0\}$ in I is compact.

which is correct?

Answer 1

Why do you have both $g$ and $f$? The answer is

$$ C_c^{\infty}(I) = \{f:I \to \mathbb F \mid \text{$f$ is infinitely differentiable with compact support} \} $$ where $\mathbb F$ could be $\mathbb R$ or $\mathbb C$. Which one will depend on the author, and the author should really specify. If they haven't, $\mathbb R$ would be my best guess. Also remember the support of a function is the closure of the set on which it is non-zero.