By Roger Godement

Features in R and C, together with the speculation of Fourier sequence, Fourier integrals and a part of that of holomorphic features, shape the focal subject of those volumes. in accordance with a direction given through the writer to giant audiences at Paris VII collage for a few years, the exposition proceeds just a little nonlinearly, mixing rigorous arithmetic skilfully with didactical and historic concerns. It units out to demonstrate the range of attainable ways to the most effects, so as to begin the reader to equipment, the underlying reasoning, and primary rules. it really is appropriate for either educating and self-study. In his general, own type, the writer emphasizes principles over calculations and, averting the condensed sort usually present in textbooks, explains those rules with out parsimony of phrases. The French variation in 4 volumes, released from 1998, has met with resounding good fortune: the 1st volumes at the moment are to be had in English.

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**Additional info for Analysis II: Differential and Integral Calculus, Fourier Series, Holomorphic Functions (Universitext)**

**Example text**

The regulated functions of Chap. II, n◦ 11, have (Chap. III, n◦ 12) both left and right limits at every point of I. In this n◦ and the following, we shall show that this property characterises them, if I is compact. The idea of the proof is very simple: the whole problem is to show that, for every r > 0, one can decompose I into a ﬁnite number of subintervals Ik such that the given function f is constant to within r on each Ik . This condition is clearly necessary if the condition (1) is to be satisﬁed; if, conversely, it is satisﬁed, and if one assumes, as one may, that the Ik are pairwise disjoint, one obtains (1) on taking ϕ to be equal to f (ξk ) on Ik , where ξk is a point chosen arbitrarily in Ik .

In particular: Corollary 2. Let f be a function deﬁned and continuous on a bounded set X ⊂ C. The following two properties are equivalent: (i) f is uniformly continuous on X; (ii) f is the restriction to X of a continuous function on the compact set X. We have just seen that (i) implies (ii). The converse implication follows from Theorem 8 since X is compact. If, for example, X =]0, 1], the function f (x) = sin(1/x) manifestly has no limit when x tends to 0; this does not prevent it from being integrable since it is continuous and bounded (Corollary to Theorem 6), but does prevent it from being uniformly continuous on X.

III, n◦ 12 (Corollary of Theorem 16). The implication (i) =⇒ (ii) is obtained, when I is compact, by observing, as at the beginning of the preceding n◦ , that for every r > 0, there exists for every x ∈ I an open interval U (x) =]x − r , x + r [ such that f is constant to within r on each of the three intervals ]x − r , x[, [x, x] and ]x, x + r [; it remains only to apply BL to the U (x) to obtain a ﬁnite number of intervals covering I and on each of which f is constant to within r; this argument also shows that f is bounded on every compact K ⊂ I.