On a complete characterization of the set of points of unbounded grade of an arbitrary surface

Abstract

Let z=f(x, y) be an arbitrary surface S, in the sense that f(x, y) is an arbitrary one-valued real function of the real variables x and y. By the grade of a segment joining two points A and B, we understand the absolute value of the tangent of the angle which AB makes with the :ry-plane. The point A = (£, rj, f ) of the surface S is said to be of bounded grade—or S is said to be of bounded grade at the point (£, rj)—if the grade of AB is bounded for all B = (x, y, z) of 5 at a sufficiently small horizontal distance [{x — %) + (y — rj)] from A. HA does not satisfy this condition, S is said to be of unbounded grade at (£, rj). It is the object of the present paper to prove the following theorem, which identifies the aggregate—for the totality of arbitrary surfaces—of sets of points of unbounded grade with the aggregate of sets of type Go.f