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!