Title: An Analytic Framework for Robust Training of Differentiable Hypothesis. Abstract: The reliability of a learning model is key to the successful deployment of machine learning in various industries. Creating a robust model, particularly one unaffected by adversarial attacks, requires a comprehensive understanding of the adversarial examples phenomenon. However, it is difficult to describe the phenomenon due to the complicated nature of the problems in machine learning. Consequently, many studies investigate the phenomenon by proposing a simplified model of how adversarial examples occur and validate it by predicting some aspect of the phenomenon. While these studies cover many different characteristics of the adversarial examples, they have not reached a holistic approach to the geometric and analytic modeling of the phenomenon. We observe the phenomenon in many applications of machine learning, and its effects seems to be independent of the choice of the hypothesis class. In this paper, we propose a formalization of robustness in learning theoretic terms and give a geometrical description of the phenomenon in analytic classifiers. We then utilize the proposal to devise a robust classification learning rule for differentiable hypothesis classes and showcase our framework on synthetic and real-world data.