With C1 convergence plus a positive lower bound on critical point spacing, maxima, minima, and saddle counts converge; with C2 convergence to a Morse function, Morse index counts converge.
de Gouveia dal Pino, Annibal Hetem, Jorge E
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Regularity Conditions for Critical Point Convergence
With C1 convergence plus a positive lower bound on critical point spacing, maxima, minima, and saddle counts converge; with C2 convergence to a Morse function, Morse index counts converge.