This paper introduces a promising alternative method for training Generative Adversarial Networks (GANs) on large-scale datasets with clear theoretical guarantees. GANs are typically learned through a minimax game between a generator and a discriminator, which is known to be empirically unstable. Previous learning paradigms have encountered mode collapse issues without a theoretical solution. To address these challenges, we propose a novel Lipschitz-constrained Functional Gradient GANs learning (Li-CFG) method to stabilize the training of GAN and provide a theoretical foundation for effectively increasing the diversity of synthetic samples by reducing the neighborhood size of the latent vector. Specifically, we demonstrate that the neighborhood size of the latent vector can be reduced by increasing the norm of the discriminator gradient, resulting in enhanced diversity of synthetic samples. To efficiently enlarge the norm of the discriminator gradient, we introduce a novel ε-centered gradient penalty that amplifies the norm of the discriminator gradient using the hyper-parameter ε. In comparison to other constraints, our method enlarging the discriminator norm, thus obtaining the smallest neighborhood size of the latent vector. Extensive experiments on benchmark datasets for image generation demonstrate the efficacy of the Li-CFG method and the ε-centered gradient penalty. The results showcase improved stability and increased diversity of synthetic samples.