What is meant by $C(\mathbb{R})$

Question

I'm homelearning calculus and trying to decide whether the following statement is true or false:

If $f \in C(\mathbb{R})$ and $\lim_ {x \rightarrow \infty} f(x) =0$, then also $\lim_ {x \rightarrow \infty} F(x)=0$, where $F$ is primitive function to $f$ on $\mathbb{R}$.

However, I'm sure what is $C(\mathbb{R})$. Could you please help me?

Thanks

Answer 1

$C(\mathbb{R})$ typically denotes the space of continuous functions $\mathbb{R}\to \mathbb{R}$. This can also be denoted by $C^0(\mathbb{R})$ in some contexts.

Answer 2

$C(\Bbb R)=\{f:\Bbb R\to\Bbb R: f\,\text{is continuous}\}$.