A question on infinite derivative.

Definition of infinite derivative: Let $\displaystyle E=\text{dom }f, c\in E$, *c* is a limit point of *E*, real-valued function *f* is said to have a infinite derivative at *c* if *f* is continuous at *c* and 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 both *f* and *g* has infinite derivative at *c*, what about the differentiability of the sum *f*+*g* at *c*? 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 at *c*=0 even if $\displaystyle f'(c)+g'(c)=(+\infty)+(-\infty)$ which is undefinable. Now I want to find an example such that $\displaystyle f'(c)=+\infty, g'(c)=-\infty$, but *f*+*g* is not differentiable at *c*, where *c* is a finite real number. I hope to find a function *F* satisfying 1)continuous at *c*, 2)has bounded derivative about *c* but except *c* and 3)$\displaystyle F'$ oscillates like $\displaystyle \sin\frac{1}{x}$ so that *F* is not differentiable at *c*. If I can find such *F*, My example can be easily constructed. But I failed to find such *F*, Can you help me find such F, or construct the example in another way? Thanks!