Abstract: In recent years, online learning has been extensively studied due to its huge application value. However, in many open environment application scenarios, the data may have new features at the current moment, and only part of the original features at the next moment are inherited. For example, in environment monitoring, with the deployment of new sensors, new features appear; when some of the old sensors are out of operation, only some of the original features of the data are retained. In this paper, such data is called streaming data with inheritably increasing and decreasing features. Traditional online learning algorithms are based on the fixed feature space, and cannot directly deal with data with inheritably increasing and decreasing features. To solve the problem, we propose online classification with feature inheritably increasing and decreasing (OFID), together with its two variants. When new features appear, the classifiers on the original features and new features are updated by combining the online passive-aggressive algorithm and the principle of structural risk minimization. When the old features disappear, the frequent-directions algorithm is used to complete the data matrix which allows the old classifier to continue to update. We theoretically analyze the performance bounds of the proposed algorithms and extensive experiments demonstrate the effectiveness of our algorithms.