The limited existence of ∝-derivatives with ∝ ≠ 1

Let: f:[0,1] → R have the property that

[Formula Matemática]

exists at X_o E [0,1), where ∝ > 0 and ∝ ≠ 1. Moore and Nashed [4] have found interesting applications for a generalization of this derivative of degree a, as it might be called.