Deep neural networks have been widely adopted to many real-world application and their reliability has been widely concerned. This paper introduces a notion of ε-weakened robustness (briefly as ε-robustness) for analyzing the reliability and some related quality issues of deep neural networks. Unlike the conventional robustness, which focuses on the ``perfect'' safe region in the absence of adversarial examples, ε-weakened robustness focuses on the region where the proportion of adversarial examples is bounded by user-specified ε. The smaller the value of ε is, the less vulnerable a neural network is to be fooled by a random perturbation. Under such robustness definition, we can give conclusive results for the regions where conventional robustness ignores. We propose an efficient testing based method with user-controllable error bounds to analyze it. The time complexity of our algorithms is polynomial in the dimension and size of the network. So, they are scalable to large networks. One of the important application of our ε-robustness is to build a robustness enhanced classifier to resist adversarial attack. Based on this theory, we design a robustness enhancement method with a good interpretability and rigours robustness guarantee. The basic idea is to resist perturbation with perturbation. Experimental results show that our robustness enhancement method can significantly improve the ability of deep models to resist adversarial attacks while maintaining the prediction performance on the original clean data. Besides, we also show the other potential value of ε-robustness in neural networks analysis.