q-differentiability in $R_q$

Question

Wokring in q-calculus where everything is defined on the set $$R_q =\{\pm q^k,k \in \mathbb{Z} \} \cup \{ 0\}$$ In which they define the q-derivative (or q-difference operator) as $$D_qf(x)=\frac{f(x)−f(qx)}{(1−q)x}, \quad x\neq 0$$ $$D_qf(x)=f'(0), \quad x=0 $$ My question is, what is meant by q-differentialbe here as the only singularity is at $x=0$ which is excluded. Then every function should be q-differentiable or not?