  The present paper studies a quantitative version of the transversality
theorem. More precisely, given a continuous function $f\in
\mathcal{C}([0,1]^d,\mathbb{R}^m)$ and a manifold $W\subset \mathbb{R}^m$ of
dimension $p$, a sharpness result on the upper quantitative estimate of the
$(d+p-m)$-dimensional Hausdorff measure of the set
$\mathcal{Z}_{W}^{f}=\left\{x\in [0,1]^d: f(x)\in W\right\}$, which was
achieved in [8], will be proved in terms of power functions.
