Metric derived numbers and continuous metric differentiability via homeomorphisms

We define the notions of unilateral metric derivatives and ``metric derived numbers'' in analogy with Dini derivatives (also referred to as ``derived numbers'') and establish their basic properties. We also prove that the set of points where a path with values in a metric space with continuous metric derivative is not ``metrically differentiable'' (in a certain strong sense) is $\sigma$-symmetrically porous and provide an example of a path for which this set is uncountable. In the second part of this paper, we study the continuous metric differentiability via a homeomorphic change of variable.