On integrability in F-spaces

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 ${\displaystyle l_p}$-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 functions which tends to x uniformly and for which the sequence of derivatives tends to y uniformly. There is also constructed a differentiable function x with ${\displaystyle x\prime \left(t_0\right)=x_0}$ for given ${\displaystyle t_0}$ and ${\displaystyle x_0}$ and x'(t) = 0 for ${\displaystyle t\ne t_0}$.