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Like our human brain, we can first learn (train) the handwriting 0 and 1. After the traing (and test) accuray is good enough, we only need to study (traing) the hardwriting 2, Instead of cleaning all of learned memory, and relearn handwriting data 0, 1, and 2 at the same time.
Can CNN do the same thing? Can CNN learn something new, but keep the previous memory? If yes, the efficiency could be high. Right now, I have to give all of data at the same time, the efficiency is very very low.
You are looking for incremental (or online) learning.
A CNN can be trained incrementally. For example, in the paper Incremental Learning of Convolutional Neural Networks, the authors propose an incremental learning algorithm (inspired by AdaBoost and Learn++, which is another incremental learning algorithm for supervised learning of neural networks) for CNNs.
However, note that incremental learning is a challenging task, given the stability-plasticity dilemma: a completely stable model, in order to keep being stable, will attempt to preserve the existing knowledge, so it will not learn new knowledge; similarly, a completely plastic model, in order to keep being plastic, it will keep forgetting previously acquired knowledge so that to learn new information.
