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We give, in terms of the Łojasiewicz inequality, a sufficient condition for C k -mappings germs of non-isolated singularity at zero to be isotopical.

Abstract. Let I, g : (Rn, 0) --+ (R , 0) be analytic functions. We will show that if 'il 1(0) = 0 and g IE ur 12 then I and g are cr-right equivalent, where (f) denote ideal generated by I and r E N.

Let f, g : ‫ޒ(‬ n , 0) → ‫,ޒ(‬ 0) be C r+1 functions, r ∈ ‫.ގ‬ We will show that if ∇f (0) = 0 and there exist a neighbourhood U of 0 ∈ ‫ޒ‬ n and a constant C > 0 such thatand for any m ∈ ‫ގ‬ n 0 such that |m| ≤ r, then there exists a C r diffeomorphism ϕ : ‫ޒ(‬ n , 0) → ‫ޒ(‬ n , 0) such that f = g • ϕ in a neighbourhood of 0 ∈ ‫ޒ‬ n .

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