Are there functions with a constant output regardless of input, or functions whose input is limited to a single number? (Cartesian coordinates)

Question

I was wondering whether a function $y=f(x)$ can be defined such that (1) its value $y$ is always constant, no matter what number substitutes $x$, (2) its argument $x$ is limited to a single number? Examples of such function definitions would be greatly appreciated.

Answer 1

If $X$ and $Y$ are any sets, then a function from $X$ to $Y$ is simply an assignment of an element of $Y$ to each element of $X$. (More formally, it is a subset of the product set $X\times Y$ that satisfies certain conditions.)

So...

For (1), we can define for example a function $f:\Bbb R\rightarrow\Bbb R$ by $f(x)=99$ for all $x\in \Bbb R$.

For (2), we can only define such a function if $X$ consists of a single element. Not very useful!

Answer 2

A function is two sets with a mapping between them. Its domain could indeed consist of a single element, much like how its output could also be a single number. Think of the identity function $f:\{1\} \to \{1\}$ where $f(1)=1$. It clearly satisfies both of the properties you give.