Re: What functions are these?

There is a very important number-theory function
that maps each natural number n
onto the number of smaller numbers coprime to it,
which is also the number of invertible elements modulo n.
Can't remember what it is called, though!

But your function is interesting too.
If n = p^r with p prime, then f(n) = r+1.
And if m and n are coprime, f(mn) = f(m) f(n)

As for your step functions, number theorists generally aren't
interested
in extending their functions to non-integers.

.

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