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The notion of the miniimum of anintegral in the calculiis of variationls is analogous to that of the minimum of a function of a real variable. The function, f(x), is thought of as continuous and it is said to have a minimum at the point x = a when, for all points of the nleighborhood of a with the one exception of a itself, the value of f(x) is greater thanf(ca); i. e., when the relation holds: /(t) >f (a), ? < Ix al ch. This is, however, only a partial description of the behavior of the function in the neighborhood of the point a, as is shown by a simple example of a discontinuous functionl for which the above relation is true. Let the function ?. (x) be defined as follows:

On a fundamental property of a minimum in the calculus of variations and the proof of a theorem of Weierstrass’s