  Let $(X,d,p)$ be a pointed metric space. A pretangent space to $X$ at $p$ is
a metric space consisting of some equivalence classes of convergent to $p$
sequences $(x_n), x_n \in X,$ whose degree of convergence is comparable with a
given scaling sequence $(r_n), r_n\downarrow 0.$ A scaling sequence $(r_n)$ is
normal if this sequence is eventually decreasing and there is $(x_n)$ such that
$\mid d(x_n,p)-r_n\mid=o(r_n)$ for $n\to\infty.$ Let
$\mathbf{\Omega_{p}^{X}(n)}$ be the set of pretangent spaces to $X$ at $p$ with
normal scaling sequences. We prove that $\mathbf{\Omega_{p}^{X}(n)}$ is
uniformly bounded if and only if $\{d(x,p): x\in X\}$ is a so-called completely
strongly porous set. It is also proved that the uniform boundedness of
$\mathbf{\Omega_{p}^{X}(n)}$ is an equivalent of the constancy of metric
derivatives of all metrically differentiable mappings on $X$ in the open balls
of a fixed radius centered at the marked points of pretangent spaces.
