Is $(-1)^x$ a Function of $x$ if domain is a set of natural numbers?

Question

Is $(-1)^x$ a Function of $x$ if domain is a set of natural numbers? If yes, what kind of function is it and how to differentiate it?

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By definition, a function is a rule that assigns to each element $x$ of the domain exactly one element called $f(x)$.

Answer 1

Definition: $f\subset A\times B$ is a function iff $\forall x\in\Bbb{R}:\exists!y\in B:(x,y)\in f$, we write $y=f(x)$ for this unique $y$.

Let $A$ be the set of natural numbers. $A=\Bbb{N}$, and let $B=\{-1,1\}$. We can define your function $f:A\rightarrow B$ as follows:

$$f(x)=\begin{cases} -1, & \text{if}\ x \ \text{is odd}\\ 1, & \text{if}\ x \ \text{is even}\\ \end{cases}$$ Trivially, every $x\in \Bbb{N}$ is either odd or even (and not both) so we have an unique value for all possible entries.

Differentiability: Because your function is discrete, not continuous, is is not differentiable.