Question I need help with determining the flaw in the proof

Question I need help with determining the flaw in the proof for the Chain Rule. First, I will list the Theorem and then I will list the proof and identify the known flaw. I need to explain the flaw and how the Chain Rule could be restated to allow this proof to be correct.

Proof:

Please show reasoning why 3rd step is flawed and how you could restate the Chain Theorem to make this proof correct. Thanks...

Solution

problem arises if f(x) is constant around the neighbourhood of c. in that case f(x) - f(c) = 0 and division by zero takes place(note that it is different from taking limx->c (f(x) - f(c)),because limit in general is not sure to go to 0 but in constant case it is always 0).The remeady is to put this additional constraint in the formulation that f(x) is not constant in the neighbourhood of c.