Business processes are omnipresent in nowadays economy: companies operate repetitively to achieve their goals, e.g., deliver goods, complete orders. The business process model is the key to understanding, managing, controlling, and verifying the operations of a company. Modeling of business processes may be a legal requirement in some market segments, e.g., financial in the European Union, and a prerequisite for certification, e.g., of the ISO-9001 standard. However, business processes naturally evolve, and continuous model adaptation is essential for rapid spot and reaction to changes in the process. The main contribution of this work is the Continuous Inductive Miner (CIM) algorithm that discovers and continuously adapts the process tree, an established representation of the process model, using the batches of event logs of the business process. CIM joins the exclusive guarantees of its two batch predecessors, the Inductive Miner (IM) and the Inductive Miner – directlyfollows-based (IMd): perfectly fit and sound models, and single-pass event log processing, respectively. CIM offers much shorter computation times in the update scenario than IM and IMd. CIM employs statistical information to work around the need to remember event logs as IM does while ensuring the perfect fit, contrary to IMd.