  We strengthen the results of \cite{A1}, consequently, we improve the claims
of \cite{A2} obtaining the best possible results. Namely, we prove that if a
subgroup $\Gamma $ of $\mathrm{Diff}_{+}(I)$ contains a free semigroup on two
generators then $\Gamma $ is not $C_0$-discrete. Using this, we extend the
H\"older's Theorem in $\mathrm{Diff}_{+}(I)$ classifying all subgroups where
every non-identity element has at most $N$ fixed points. In addition, we obtain
a non-discreteness result in a subclass of homeomorghisms which allows to
extend the classification result to all subgroups of $\mathrm{Homeo}_{+}(I)$
where every non-identity element has at most $N$ fixed points.
