Definition of infinite derivative: Let $\displaystyle E=\text{dom }f, c\in E$,cis a limit point ofE, real-valued functionfis said to have a infinite derivative atciffis continuous atcand the limit $\displaystyle \lim\limits_{x\to c}\frac{f(x)-f(c)}{x - c}$ is $\displaystyle +\infty$ or $\displaystyle -\infty$. We write in this case $\displaystyle f'(c)=+\infty$ or $\displaystyle f'(c)=-\infty$.

For finitely differentiable functions it is well-known that $\displaystyle (f+g)'=f'+g'$. But if bothfandghas infinite derivative atc, what about the differentiability of the sumf+gatc? It may be still differentiable, e.g. $\displaystyle f=x^{1/3}$ and $\displaystyle g=-x^{1/3}$,f+g=0 so (f+g)'=0 atc=0 even if $\displaystyle f'(c)+g'(c)=(+\infty)+(-\infty)$ which is undefinable. Now I want to find an example such that $\displaystyle f'(c)=+\infty, g'(c)=-\infty$, butf+gis not differentiable atc, wherecis a finite real number. I hope to find a functionFsatisfying 1)continuous atc, 2)has bounded derivative aboutcbut exceptcand 3)$\displaystyle F'$ oscillates like $\displaystyle \sin\frac{1}{x}$ so thatFis not differentiable atc. If I can find suchF, My example can be easily constructed. But I failed to find suchF, Can you help me find such F, or construct the example in another way? Thanks!