To construct a nonlinear fractional-order neural network reflecting the complex environment of the real world, this paper considers the common factors such as uncertainties, perturbations, and delays that affect the stability of the network system. In particular, not only does the activation function include multiple time delays, but the memristive connection weights also consider transmission delays. Stemming from the characteristics of neural networks, two different types of discontinuous controllers with state information and sign functions are devised to effectuate network synchronization objectives. Combining the finite-time convergence criterion and the theory of fractional-order calculus, Mittag-Leffler synchronization conditions for fractional-order multi-delayed memristive neural networks (FMMNNs) are derived, and the upper bound of the setting time can be confirmed. Unlike previous jobs, this article focuses on applying different inequality techniques in the synchronous analysis process, rather than comparison principles to manage the multi-delay effects. In addition, this study removes the restrictive requirement that the activation function has a zero value at the switching jumps, and the discontinuous control protocol in this paper makes the networks achieve synchronization over a finite time, with some advantages in terms of the convergence speed.