Regularity Regions of Differential Algebraic Equations

General nonlinear differential algebraic equations with properly involved derivative are characterized - without the computation of solutions, and without the use of derivative array functions. A regularity region is given by means of special sequences of matrix functions which involve certain admissible projector functions. This concept is accompanied by several rank conditions the failure of which indicates critical points. A regularity region consists of regular points, and it is bordered by critical points. An index, more precisely, the tractability index is attached to each regularity region. Solutions may cross the borders of regularity regions, but then they may show a singular behavior.