# are the sets countable?

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

• You know the Intermediate Value Theorem? – Lord Shark the Unknown Jan 18 '18 at 19:47
• What do you mean "with values in $\mathbb Z$"? DO you m fnction $\mathbb R \to \mathbb Z$? Why not right it as such. The way you stated your reasoning for 1) isn't yet correct but on the right track. $\{f:\mathbb N\to \{0,1\}\} \subset \{f:\mathbb N\to \mathbb Z\} \subset \{f:\mathbb R\to \mathbb Z\}$. And now you can state it. 2) has a trick in that continuous function can only be constant. – fleablood Jan 18 '18 at 19:53
• thank you both. @LordSharktheUnknown yes I know this theorem – user472520 Jan 18 '18 at 19:57
• ah, I can use the intermediate value theorem to see that continuous functions $\mathbb{R}\to\mathbb{Z}$ must be constant, right – user472520 Jan 18 '18 at 20:09

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?
• Then, the cardinality of continuous functions $\mathbb{R}\to\mathbb{Z}$ agrees with the cardinality of $\mathbb{Z}$, and is therefore countable. Is it correct? – user472520 Jan 18 '18 at 20:00