For an operator to be a linear function, they must meet two criterias:

1. Additivity, i.e. addition is preserved
2. Homogeneity of degree 1, i.e. scalar multiplication is preserved

In other words, the following are satisfied:

• $f(x+y) = f(x) + f(y)$
• $f(cx) = c f(x)$

for the function $f$ of interest. It turns out that discrete derivatives satify these conditions.

Additivity

Homogeneity of degree 1

So we can conclude that the discrete derivative as an operation is a linear transformation in the function space.