Countability - Constant Functions

Question

I am learning about countability. I know about diagonalization and I am confused about constant functions and whether or not they are countable.

A constant function in my case would be:

$f(0) = 1,$ $f(1) = 1,$ $f(2) = 1,$ $f(3) = 1,$ $f(1000) = 1.$

So for $f(0) \dots f(9)$ I would have the sequence:

$(1, 1, 1, 1, 1, 1, 1, 1, 1, 1)$

How would I show this is countable or not countable?

I tried diagonalization with:

$1, \, 1, \,1, \,1, \,1, \,1, \,1, \,1, \,1, \,1$ and

$2, \, 2, \,2, \,2, \,2, \,2, \,2, \,2, \,2, \,2$

and created a new sequence: $2, 3 \dots$ that is not on the original list.

This states that constant functions are not countable, my my intution tells me that they should be countable.

Answer 1

The set $\{ f \colon \mathbb{N} \to \mathbb{N} \colon f $ is constant $\}$ is indeed countable, you have the bijection

$n \longmapsto f_n$ where $f_n(x) = n$ for all $x \in \mathbb{N}$

Answer 2

Define $f: \{All constant functions N to N\}->N$ by f(g)=g(0).

Note that $f$ is bijective, therefore the set is countable.