By R.B. Burckel

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**Additional info for An Introduction to Classical Complex Analysis: Vol. 1**

**Sample text**

32. Set R(x, r) = {y E IR": \\x - Y\\ r} $ S = {(x,r)EIR" x [O,oo):d(x) $ r} (x, r) E S. p is continuous on the closed set S. p(x, r)dr, x E IR"\C. 38 Curves, Connectedness and Convexity Prove that 4> is continuous and satisfies in addition CP(Xk) -+ f(x) whenever x E C. Finally define the desired extension by Xk E IRn\c and X k -+ F={f inC cp in IRn\c. 45 (BORSUK [1937]) Let A be a compact subset of a closed subset B of IRn. Let f: A -+ q{O} and g: B -+ q{O} be continuous and suppose there exists a continuous function h: A x [0, 1]-+ q{O} so that hex, 0) = f(X)} hex, 1) = g(x) XE A • Then f has a continuous extension F: B -+ q{O}.

28 are examples; here is another: Theorem (POMPEIU [1905a], LOOMAN [1925], RIDDER [1930a]) If/is continuous in U, lim" .... ol(f(z + h) - /(z))/hl isfinite for all but countably many z in U and f' exists almost everywhere in U, then / is holomorphic in U. For details of these results, which are heavily real-variable in nature, the reader may consult MENCHOFF [1936], pp. 195-201 of SAKS [1937] or the exhaustive (but terse) treatise of TROKHIMCHUK [1964]. For a somewhat more elementary account of somewhat weaker results in this direction, see MEIER [1951], [1960] and ARSOVE [1955].

There is, however, some esthetic loss in defining the length of (a smooth) y to be Iy'l- One ought to call a curve y: [a, b] -+ C rectifiable if there is a finite M such that the length 2:7=1 ly(1t) - y(1f-l)1 of the polygon [,,(/ 0 ), ••• , ,,(t,,)] inscribed in y is not greater than M whatever be the points a ~ 10 ~ 11 ~ ••• ~ t" ~ b. In such cases call the least upper bound of the lengths of these inscribed polygons the length of y. It is then a theorem (and an easy one, see RUDIN [1976], p.