What nontrivial operations exhibit $\text{op}(f(x)) + \text{op}(f(x+1)) = \text{op}(f(x) + f(x+1))$?
For example, I know that summation, integration, and their inverses all exhibit this property. To further illustrate this, we can take summation:
$$\sum_x{f(x)} + \sum_x{f(x+1)} = \sum_x{(f(x) + f(x+1))}$$
Or integration:
$$\int{f(x)dx} + \int{f(x+1)dx} = \int{(f(x) + f(x+1))dx}$$
So my question is: What other operations/functions have this property? Can we somehow classify them all? Also, does this property have a name?
This' neat!
These types of operations are typically found under the notions of homomorphism, ie ``structure preserving mapping''.
More precisely, suppose we have a set $S$ and a binary operation on it $\oplus:S \times S \to S$, then we call $(S,\oplus)$ a magma. (If this operation has a unit and is invertible we obtain the notion of group.)
Now suppose we have two magmas $(S,\oplus_S)$ and $(T,\oplus_T)$ then a structure-preserving mapping ---or `magma homomorphism'--- would be a map $$h : S \to T$$ such that $$h(a \oplus_S b)=h(a) \oplus_T h(b), \;\;\;\text{for any $a,b$ in $S$ }$$
In particular, if we take $h,a,b$ to be $op, f(x), f(x+1)$, respectively, then we obtain your specific instance :)
Another example is to consider the magmas $(\mathbb{R},+)$ and $(\mathbb{R},\times)$, then the exponential operation $r \mapsto \exp(r)$ is a homomorphism since $\exp(a+b) = \exp(a) \times \exp(b)$.
Hope this helps!