I found out from here that $C(a, b)$ means the space of all continuous functions on the interval $(a, b)$.

I also searched the wikipedia page about differentiability classes but didn’t see that notation being used.

I stumbled upon this notation in a lecture on ordinary differential equations. A similar one I found is $C(J, mathbb{R}^star)$

The context for the notation in the question is $g in C(J, mathbb{R}^2)$ and it is said to be a space