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A second-order equation can have singular sets of first and second type, $S\_1$ and $S\_2$ (see the introduction), where the integral curve $x(y)$ does not exist in the ordinary sense but where it can be extended by using the first integral [1--5]. Denote by $Y$ the Cartesian axis $y=\backslash nomathbreak\; 0$. If the function $x(y)$ has a derivative at a point of local extremum of this function, then this point belongs to $S\_1\backslash cup\; Y$. The extrema at which $y\text{'}(x)$ does not exist can be placed on $S\_2$. In [5--8], the stability and instability of extrema on $S\_1\backslash cup\; S\_2$ under small perturbations of the equation were considered, and the stability of the mutual arrangement of the maxima and minima of $x(y)$ on the singular set was studied (locally as a rule, i.e., in small neighborhoods of singular points). In the present paper, sufficient conditions for the preservation of type of a local extremum on the finite part of $S\_1$ or $S\_2$ are found for the case in which the perturbation on all of this part does not exceed some explicitly indicated quantity which is the same on the entire singular set. [ABSTRACT FROM AUTHOR]