Porosity and typical properties of real-valued continuous functions (Q920210)

Let C be the space of all real-valued continuous functions on [0,1]. We say that nearly all elements of C have a certain property if those not enjoying it form a \(\sigma\)-porous set. The function \(f\in C\) is said to be of monotonic type at x if there is a real number c such that the function \(t\to f(t)-ct\) is monotonic at x. It is proven: 1) Nearly all elements of C are of nonmonotonic type at any x. 2) For nearly all elements f of C at each point \(x\in [0,1):\) \(D^+f(x)=\infty\) or \(D_+f(x)=-\infty\), and at each point \(x\in (0,1]:\) \(D^-f(x)=\infty\) or \(D_ -f(x)=-\infty\). 3) For nearly all elements f of C at most points \(x\in [0,1]:\) \(D_ -f(x)=D_+f(x)=-\infty\) and \(D^- f(x)=D^+f(x)=\infty\).