A note on a paper by D.K.R. Babajee and M.Z. Dauhoo

A counterexample is provided in this short note to show that some of local convergence theorems established in [D.K.R. Babajee, M.Z. Dauhoo, An analysis of the properties of the variants of Newton's method with third order convergence, Appl. Math. Comput. 183 (2006) 659-684] are not always true. Some mistakes in the proofs of these theorems are pointed out.