No I did not do what you wrote.

Here's what I did: I defined what a function is, then I described how a unix program fits the definition of what a function is...

In short a unix program with stdin as domain and stdout as a codomain is a function but a function is not necessarily a unix program.

I think what's actually going on is you didn't read what I wrote very carefully. You sort of just skimmed over it. I can't blame you, it is rather long and detailed...

But if you want to have a meaningful discussion you need to read it and ask questions about things you don't understand.