Remarks on Frechet differentiability of pointwise Lipschitz, cone-monotone and quasiconvex functions

We present some consequences of a deep result of J. Lindenstrauss and D. Preiss on F-almost everywhere Frechet differentiability of Lipschitz functions on c(0) (and similar Banach spaces). For example, in these spaces, every continuous real function is Frechet differentiable at Gamma-almost every x at which it is Gateaux differentiable. Another interesting consequences say that both cone-monotone functions and continuous quasiconvex functions on these spaces are Gamma-almost everywhere Frechet differentiable. In the proofs we use a general observation that each version of the Rademacher theorem for real functions on Banach spaces (i.e., a result on a.e. Frechet or Gateaux differentiability of Lipschitz functions) easily implies by a method of J. Maly a corresponding version of the Stepanov theorem (on a.e. differentiability of pointwise Lipschitz functions). Using the method of separable reduction, we extend some results to several non separable spaces.