On Hadamard differentiability
Let be a bounded domain of Lyapunov and a holomorphic function in the cylinder and continuous on . If for each fixed in some set , with positive Lebesgue measure , the function of can be continued to a function holomorphic on the whole plane with the exception of some finite number (polar set) of singularities, then can be holomorphically continued to , where is some analytic (closed pluripolar) subset of .
Some usual and unusual properties of the Riemann integral for functions x : [a,b] → X where X is an F-space are investigated. In particular, a continuous integrable -valued function (0 < p < 1) with non-differentiable integral function is constructed. For some class of quasi-Banach spaces X it is proved that the set of all X-valued functions with zero derivative is dense in the space of all continuous functions, and for any two continuous functions x and y there is a sequence of differentiable...