= Solution
For the <phase-space dilation generator> $D=q\cdot p$,
$$
\frac{\partial D}{\partial p_i}=q_i,
\qquad
\frac{\partial D}{\partial q_i}=p_i.
$$
Therefore the condition $\{f,D\}=0$ is the first-order equation
$$
\boxed{q\cdot\nabla_qf-p\cdot\nabla_pf=0.}
$$
If $f(\lambda q,\lambda^{-1}p)=f(q,p)$ for every positive real $\lambda$, differentiate with respect to $\lambda$ at $\lambda=1$. The <chain rule> gives exactly
$$
q\cdot\nabla_qf-p\cdot\nabla_pf=0,
$$
so every smooth scale-invariant $f$ satisfies the constraint. The same conclusion for negative $\lambda$ follows wherever the stated invariance is defined; differentiation only needs the positive component containing $\lambda=1$.
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