By V. P. Havin, S.V. Hruscev, N.K. Nikol'skii

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In Example B, f(x) is discontinuous at x = 0 on two counts: f(0) is undefined, and the limit involved does not exist. No choice of definition for f(0) could make f(x) continuous at x = 0. If in Definition 1 “x → a” is replaced by “x → (“x → a−”), f(x) is said to be continuous on the right (left) at x = a. Thus, in Example C, f(x) is continuous on the right at x = 0 if f(0) = 0. We say that f(x) ∈ C, a < x < b, ⇔f(x) ∈ C at each x of the interval a < x < b. 2 DERIVATIVES We now introduce further classes of functions, those which have derivatives of certain orders.

Certain theorems have been sharpened, where this could be done without too much sacrifice of simplicity. For example, the class of differentiable functions of several variables has been interpolated between C and C1. The treatment of Stieltjes integrals has been altered somewhat with the purpose of making it more useful to the student who is not very familiar with the basic facts about the Riemann integral. In fact, such a student may, if he wishes, correct his deficiency without studying the Stieltjes integral by concentrating on sections 1, 6, and 7 of Chapter 5.

However, it is arranged in such a way that it may also be used to advantage by students with somewhat less preparation. The reader is expected to have considerable skill in the manipulations of elementary calculus, but it is not assumed that he will be very familiar with the theoretic side of the subject, Consequently, the book emphasizes first the type of manipulative problem the student has been accustomed to and gradually changes to more theoretic problems. In fact, the same sort of crescendo appears within the chapters themselves.

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