Is $f(x) = \infty$ a function?

Question

Recently, while solving a problem where a certain set of functions $f:\mathbb Z^+ \rightarrow \mathbb Z^+$ had to be found given a number of conditions, I noticed that $f(n)=\lim_{a\to+\infty} a$, where $n\in \mathbb Z^+$, was a solution.

My question is simply whether $f(n)=\lim_{a\to+\infty} a$ really can be classified as a function. If not, why not, and if yes, does the condition $f:\mathbb Z^+ \rightarrow \mathbb Z^+$ still hold?

Since $\infty\not\in\mathbb Z^+$, it is not a function of the sort you were seeking. — Thomas Andrews, Sep 21 '14 at 17:03
@Hakim The delta function is also not a function, despite its name. — Thomas Andrews, Sep 21 '14 at 17:04
The set $\mathbb{Z}^+$ has infinitely many elements. It has no "infinite" elements, whatever that may mean. — André Nicolas, Sep 21 '14 at 17:04
@ThomasAndrews Yeah, I was just suggesting him some further reading material. — Hakim, Sep 21 '14 at 17:50
Is it really related? Given his question, and your lack of verbosity, might you think that comment is a bit misleading? @Hakim — Thomas Andrews, Sep 21 '14 at 18:28
You're right @Thomas, comment deleted! I'm sorry for this extreme lack of clarity and the confusion engendered. — Hakim, Sep 21 '14 at 18:31

score 5 · Accepted Answer · answered Sep 21 '14 at 17:09

We say $f:A\to B$ is a function if, for any $a\in A$ there exists exactly one $b\in B$ such that $f(a)=b$.

Since $\infty$ is not an element of $\mathbb Z^+$, if you want $f:\mathbb Z^+\to \mathbb Z^+$, you can't have $f(n)=\infty$ for any positive integer $n$.

Is $f(x) = \infty$ a function?

1 Answers1