  The framework of a new scale invariant analysis on a Cantor set $C\subset $
$% I=[0,1] $, presented originally in {\it S. Raut and D. P. Datta, Fractals,
17, 45-52, (2009)}, is clarified and extended further. For an arbitrarily small
$\varepsilon >0$, elements $\tilde{x}$ in $I\backslash C$ satisfying
$0<\tilde{x}<\varepsilon <x, x\in C $ together with an inversion rule are
called relative infinitesimals relative to the scale $\varepsilon$. A
non-archimedean absolute value $v(% \tilde{x})=\log_{\varepsilon
^{-1}}\frac{\varepsilon}{\tilde{x}}, \varepsilon \to 0$ is assigned to each
such infinitesimal which is then shown to induce a non-archimedean structure in
the full Cantor set $C$. A valued measure constructed using the new absolute
value is shown to give rise to the finite Hausdorff measure of the set. The
definition of differentiability on $% C$ in the non-archimedean sense is
introduced. The associated Cantor function is shown to relate to the valuation
on $C$ which is then reinterpreated as a locally constant function in the
extended non-archimedean space. The definitions and the constructions are
verified explicitly on a Cantor set which is defined recursively from $I$
deleting $q$ number of open intervals each of length $\frac{1}{r}$ leaving out
$p$ numbers of closed intervals so that $p+q=r.$
