In this paper, a detrimental mathematical mistake is pointed out in the proof of \textit{Theorem 1} presented in the paper\textit{ [Generalization of the gradient method with fractional order gradient direction, J. Franklin Inst., 357 (2020) 2514-2532]}. It is highlighted that the way the authors prove the convergence of the fractional extreme points of a real valued function to its integer order extreme points lacks correct and valid mathematical argument. Rest of the theorems contained in the paper are mostly announced without any proof relaying on that of Theorem 1.