Can I define a function as the set of these points? Can I define a function as the set of these points as $k$ goes to infinity?

$$ \lim_{k\to \infty}\bigg (\frac{k}{k-n},\frac{k-n}{k}\bigg)$$

Where $n$ is the ordered set of natural numbers less than $k$ from least to greatest. $n=1,2,3,...,k-1.$ A point is plotted for each $n.$

For example for $k=10$ it would look like:

$$ \bigg (\frac{10}{10-n},\frac{10-n}{10}\bigg) $$

and there would be $9$ points because $n=1,2,3,4,...,9.$
A function consists of two sets (the domain and the range) and a graph between them. 
I think you have a certain relation between the points but the concept of a limit has nothing to do with this.
Examine the definition of a function until it is clear for you.
The function $f(x)=1-\frac{1}{x}$ passes through all the desired points and therefore is the correct one.