Let e[J(n, 1) be the set of analytic function germs : (RTh, O)-(R, 0). In [7]-[12], T. C. Kuo has introduced some notions of blow-analycity as desirable (natural) equivalence relations for elements of 1), and has given important results to construct blow-analytic theory. Stimulated by his work, several singularists started on studying blow-analycity and introduced notions similar to but different from those of Kuo's ([2], [3], [5], [6], [16], [18]). These blow-analytic equivalences are slightly weaker than bianalycity, and much stronger than homeomorphism. In this note, we introduce the notion of strong C°-equivalence as one of blow-analytic equivalences. Roughly speaking, it is a C°-equivalence which preserves the tangency of analytic arcs at OERn. It seems that this equivalence is not so strong. In fact, this is weaker than some other blow-analytic equivalences. Our purposes in this paper are to formulate two conditions which imply strong C°-equivalence and to show the BrianconSpeder family ([1]) and the Oka family ([15]) are not strongly C°-trivial. In the complex case, the Briancon-Speder family is well-known as an example that topological triviality does not imply the Whitney regularity, in other words, the Milnor number constancy does not imply p*-constancy. The Oka family also is p-constant but not p*-constant. Both families have a weak simultaneous resolution, but have no strong simultaneous resolution (see [13], [14], [15], [17]). In the real case, however, the families are topologically trivial, but not even strongly C°-trivial. The author would like to thank Professors T. Fukui, T. C. Kuo, P. Milman, M. Oka and M. Shiota for useful conversations and suggestions. The author also would like to thank the University of Sydney for its support during the time this work was in process.