On an example of a function with a derivative which does not have a third order symmetric Riemann derivative anywhere (Q418221)

A differentiable function \(F\) which does not have the third order symmetric Riemann derivative at any point is constructed.NEWLINENEWLINELet \(b\in (0,1)\) and let us define a function \(g\) as NEWLINE\[NEWLINE g(y)=\frac{1}{12^3}-\frac{18y}{b-y}-\frac{4yb}{1-yb}NEWLINE\]NEWLINE for \(y\in [0,b)\). Then \(g\) is continuous on \([0,b)\) and \(g(0)>0\). Thus there is an \(a\in (0,b)\) such that \(g(a)>0\). Further let us define a function \(f_a :\mathbb{R} \rightarrow \mathbb{R} \) as followsNEWLINENEWLINE(i) \(f_a\) is periodic of period \(13a\),NEWLINENEWLINE(ii) NEWLINE\[NEWLINE f_a(x)= \begin{cases} -x^3, & x\in [0,a) \cr -x^3+4(x-a)^3, & x\in [a,2a) \cr -x^3+4(x-a)^3-6(x-2a)^3, & x\in [2a,3a) \cr (x-4a)^3, & x\in [3a,4a) \cr 0, & x\in [4a,6.5a) \cr \end{cases} NEWLINE\]NEWLINENEWLINENEWLINE(iii) \(f_a(6.5a+x)=-f_a(6.5a-x)\) if \(x\in [0,6.5a)\).NEWLINENEWLINELet \(G_{a,b}=\frac{b}{a^2} f_a\) and NEWLINE\[NEWLINE F=\sum_{n=1}^\infty G_{a^n,b^n}. NEWLINE\]NEWLINE Then \(F'\) exists on \(\mathbb{R} \) and NEWLINE\[NEWLINE \overline{SRD}^3F(x)=\limsup_{h\searrow 0} \frac{F(x+3h)-3F(x+h)+3F(x-h)-F(x-3h)}{(2h)^3}=+\inftyNEWLINE\]NEWLINE and NEWLINE\[NEWLINE \underline{SRD}^3F(x)=\liminf_{k\searrow 0} \frac{F(x+3k)-3F(x+k)+3F(x-k)-F(x-3k)}{(2k)^3}=-\infty.NEWLINE\]