We study Newton's method for determining the solution o f f ( x ) = 0 w h e n f ( x ) is required only to be continuous and piecewise continuously differentiable in some sphere about the initial iterate, x <m. First an existence, uniqueness and convergence theorem is obtained employing the modulus of continuity of the first derivative, fx(x). Under the more explicit assumption of H6lder continuity several other such results are obtained, some of which extend results of Kantorovich and Akilov [1… CONTINUE READING