Determine if the following sets are countable or not:
(1) The set of functions $\mathbb{R}\to\mathbb{R}$ with values in $\mathbb{Z}$
(2) The set of continuous functions $\mathbb{R}\to\mathbb{R}$ with values in $\mathbb{Z}$
My guess is that (1) is not countable and I think the reason is the following: I know that the set of all functions $\mathbb{N}\to \{0,1\}$ is not countable. Since the cardinality of all functions $\mathbb{N}\to \{0,1\}$ is less or equal than the cardinality of all functions $\mathbb{R}\to\mathbb{R}$ with values in $\mathbb{Z}$, the latter set is not countable. Is it correct?
I have no idea how to do (2).. I appreciate any help and hint. Thank you
Hint for $(1)$:
Is the set of functions $\Bbb N\to\{0,1\}$ countable? What do you think about $\Bbb R\to\Bbb Z$?
Hint for $(2)$:
A continuous function $\Bbb R\to \Bbb Z$ is constant. How many integer constants are there?