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These give (32) and (33), respectively, and the Corollary 11 is proved. We could again restrict ourselves, for instance, to real or nonsingular matrices. 9Nn are groupoids with neutral elements and G an abelian group. Some further 'tricks' are needed if these contain no neutral element 0, see Aczel 1964b, 1975; Taylor 1980; Kuczma 1972; and Szymiczek 1973. We first show that a result similar to Theorem 9 also holds for abelian semigroups which do not necessarily contain a neutral element (Aczel 1964&).

Aczel 1967; Aczel-Fischer 1968; Dhombres 1979, pp. ) 5. Let /:R->R be an additive function {f{x + y)=f(x)+f(y) for all x,yeR). Suppose that m > 1 is a given integer and that f(xm) =f(x)m holds for all x > 0 such that/(x) > 0. Does it follow that/is continuous, and what is its general form? 6. Same question as in the previous exercise but with the equation f(xm) =f(x)m valid for all x # 0 such that /(x) # 0 and, when m is an integer (positive or negative), different from 1 and 0. 7. Same question as in Exercise 5, but with the equation |/(x m )| — f(x)m valid for all x # 0,/(x) ¥"0,171 being an even integer different from 0.

Aczel 1966c, pp. ) Prove that all functions / : R -+ U such that: (a) / is of class C 1 (has everywhere a continuous derivative); (b) J« 0 0 /(x)dx=l; (c) for every choice of xux2 and x 3 , the function g(x)=f(xl — x)/(x 2 —x)/(x 3 —x) has its maximum at x = (xx + x 2 + x 3 )/3 are given by /(x) - l/((2;:)1/2(7)exp( - x2/2cr2). ) 8. (Aczel 1966c, pp. ) Prove that the general solution / : R -• U of the functional equation f(x + y) = ax'f(x)f(y) (x,yeU), where a > 0 is a constant, is given by where g is an arbitrary solution of Cauchy's exponential equation.

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