Let $G$ be the group of all the maps from closed interval $[0,1]$ to $\mathbb{Z}$. The subgroup $H= \left \{ f \in G :f(0)=0 \right \}$
Then
$1)$ $H$ is countable
$2)$ $H$ is uncountable
$3)$ $H$ has countable index
$4)$ $H$ has uncountable index
Solution I tried- In this question he is asking about maps not for functions. The number of maps form $[0,1]$ to $\mathbb{Z}$ must be more than $\aleph_0^\mathfrak{c}$. Now I am confused here, how to proceed further because, I have no idea what is $\mathfrak{c}$ times $\aleph_0$. please give me a hint so that I can solve this further.
Thank you.