Continual Learning, also known as Lifelong Learning, aims to continually learn from new data as it becomes available. While prior research on continual learning in automatic speech recognition has focused on the adaptation of models across multiple different speech recognition tasks, in this paper we proposed an experimental setting for \textit{online continual learning} for automatic speech recognition of a single task. Specifically focusing on the case where additional training data for the same task becomes available incrementally over time, we demonstrate the effectiveness of performing incremental model updates to end-to-end speech recognition models with an online Gradient Episodic Memory (GEM) method. Moreover, we show that with online continual learning and a selective sampling strategy, we can maintain an accuracy that is similar to retraining a model from scratch while requiring significantly lower computation costs. We have also verified our method with self-supervised learning (SSL) features.