An additive function $f(n)$ is said to be totally/completely additive if
$$f(ab) = f(a) + f(b)$$
holds for all positive integers $a$ and $b$.
Additionally, if $f$ is a totally/completely additive function, then $f(1) = 0$.
On the other hand, a multiplicative function $g(n)$ is said to be totally/completely multiplicative if
$$g(ab) = g(a)g(b)$$
holds for all positive integers $a$ and $b$.
Additionally, if g is a totally/completely multiplicative function, then $g(1) = 1$.
So now suppose there is a function $h(n)$ such that
$$h(n) = \frac{1}{2}[f(n) + g(n)]$$
where $f$ is totally additive and $g$ is totally multiplicative. (I removed the erroneous assumption on the function $h(n)$, thank you Gerry for pointing that out!)
Then
$$h(1) = \frac{1}{2}(f(1) + g(1)) = \frac{1}{2}$$
Now, let
$$i(n) = h(n + 1)$$
so that
$$i(0) = h(1) = \frac{1}{2}$$
My question is:
How do we then characterize such number-theoretic functions $i(n)$?
