Definition of Analytic Functions in Complex Analysis

In Mathematics of Classical and Quantum Mechanics by Byron and Fuller, they state that "Some authors (never mathematicians) define an analytic function as a differentiable function with a continuous derivative." ..."But this is a mathematical fraud of cosmic proportions.. "

Their main point is that you don't have to assume continuity of the first derivatives of an analytic function to prove Cauchy's Integral Theorem if you use the Goursat approach, yet I thought that really IS how an analytic function is defined, i.e. that a function of a complex variable is analytic within a region S if it is differentiable within and on the boundary of S.