A question on infinite derivative.

Definition of infinite derivative: Let , *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 is or . We write in this case or .

For finitely differentiable functions it is well-known that . 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. and , *f*+*g*=0 so (*f*+*g*)'=0 at *c*=0 even if which is undefinable. Now I want to find an example such that , 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) oscillates like 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!