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Abstract This article reviews current approaches to the calculation, analysis, synthesis and optimization under uncertainty. Studying uncertain systems is formulated as problems of the calculation, analysis and synthesis of various non-deterministic functions with parameters that serve as the relevant characteristics of these systems. All these problems are much more difficult their deterministic counterparts which should be solved in the study of systems with deterministic (exactly known) parameters. Complexity is due to the fact that the non-deterministic algebra is more complicated then algebra of deterministic numbers. The article stated and described in detail the problem of calculating and analyzing the behavior of a function which is given up to a range of values. To solve this problem, the algorithm of determination is presented. This algorithm reduces the problem to the two same – for the lower and upper boundary functions of the original incompletely defined function. In this algorithm author uses interval mathematics and interval-differential calculus. The different types of possible behavior of interval functions are highlighted (consistency, increase, decrease, expansion, contraction) and various types of extreme points of such functions (for example, the maximum point, a minimum point, the point of maximum expansion, the point of minimum extension) are shown. Theorems that allow you to define areas of different behavior of interval functions and points with different types of extreme are proved. The work of the proposed algorithm of determination for analyzing the behavior of interval functions is considered in detail. Operation of algorithm is illustrated by concrete example. References Фихтенгольц Г. М. Курс дифференциального и интегрального исчисления. Т. 1 / Г. М. Фихтенгольц. – М. : Физматлит, 2001. – 616 с. 2. Левин В. И. Интервальная производная и начала недетерминистского дифференциального исчисления / В. И. Левин // Онтология проектирования. – 2013. – № 4. – С. 72–84. 3. Wiener N. Extrapolation, Interpolation and Smoothing of Stationary Time Series / N. Wiener. – N.Y.: Technology Press and Wiley, 1949. – 180 p. (рассекреченный отчет 1942 года). 4. Колмогоров А. Н. Интерполирование и экстраполирование стационарных случайных последовательностей / А. Н. Колмогоров // Известия АН СССР. Математика. – 1941. – № 5. – С. 3–14. 5. Налимов В. В. Статистические методы планирования экстремальных экспериментов / В. В. Налимов, Н. А. Чернова. – М. : Наука, 1965. – 340 с. 6. Налимов В. В. Теория эксперимента / В. В. Налимов, Н. А. Чернова. – М. : Наука, 1971. – 320 с. 7. Заде Л. А. Понятие лингвистической переменной и его применение к принятию приближенных решений / Л. А. Заде. – М. : Мир, 1976. – 176 с. 8. Нариньяни А. С. Недоопределенность в системе представления и обработки знаний / А. С. Нариньяни // Известия АН СССР. Техническая кибернетика. – 1986. – № 5. – С. 17–25. 9. Hyvonen E. Constraint Reasoning Based on Interval Arithmetic: the Tolerance Propagation Approach / E. Hyvonen // Artificial Intelligence. – 1992. Vol. 58. – P. 19.

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