For an operator to be a linear function, they must meet two criterias:
- Additivity, i.e. addition is preserved
- Homogeneity of degree 1, i.e. scalar multiplication is preserved
In other words, the following are satisfied:
for the function of interest. It turns out that discrete derivatives satify these conditions.
Homogeneity of degree 1
So we can conclude that the discrete derivative as an operation is a linear transformation in the function space.