Definition of infinite derivative: Let ,cis a limit point ofE, real-valued functionfis said to have a infinite derivative atciffis continuous atcand the limit is or . We write in this case or .

For finitely differentiable functions it is well-known that . But if bothfandghas infinite derivative atc, what about the differentiability of the sumf+gatc? It may be still differentiable, e.g. and ,f+g=0 so (f+g)'=0 atc=0 even if which is undefinable. Now I want to find an example such that , 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) oscillates like 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!