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Let ∆(a1, a2, · · · , an) be an n-dimensional real simplex with vertices at (a1, 0, · · · , 0), (0, a2, · · · , 0), · · · , (0, 0, · · · , an). Let P(a1,a2,··· ,an) be the number of positive integral points lying in ∆(a1, a2, · · · , an). In this paper we prove that n!P(a1,a2,··· ,an) ≤ (a1 − 1)(a2 − 1) · · · (an − 1). As a consequence we have proved the Durfee conjecture for isolated weighted homogeneous singularities: n!pg ≤ μ, where pg and μ are the geometric genus and Milnor number of the singularity, respectively.

An upper estimate of integral points in real simplices with an application to singularity theory