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We present a series of C^2-Morse functions on the Deligne-Mumfordcompactification M_{g,n} bar of the moduli space of genus g Riemann/hyperbolicsurfaces with n punctures. This series of functions converges to the systolefunction, which is topologically Morse. We will show that the critical pointsof our functions approach those of systole sublinearly, stratum-wise, and withthe same indices.

C^2-Morse Functions on M_{g,n} bar