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Given a real function f on an interval [a, b] satisfying mild regularity conditions, we determine the number of zeros of f by evaluating a certain integral. The integrand depends on f, f′ and f″. In particular, by approximating the integral with the trapezoidal rule on a fine enough grid, we can compute the number of zeros of f by evaluating finitely many values of f, f′ and f″. A variant of the integral even allows to determine the number of the zeros broken down by their multiplicity.

, given ί(ξ) for ξ 6 [0, ξο] one can uniquely find function c(y) inside the layer ye[0 , / ( f (eb) ] . If d{y) > 0 for all у however the second logarithmic derivative does not satisfy to the condition (logc(y))" < 0, then function ξ(ρ) may not be a strongly monotonic function and therefore ί(ξ) may be a multi-valued function. Different situations may occur then. For example, if function ξ'(ρ) has only finite number of zeroes on an interval [pi,<fo]> then function ί(ξ) consists of finite number of single-valued smooth branches with cusp points corresponding to