  A wide variety of methods have been developed to enable lifelong learning in
conventional deep neural networks. However, to succeed, these methods require a
`batch' of samples to be available and visited multiple times during training.
While this works well in a static setting, these methods continue to suffer in
a more realistic situation where data arrives in \emph{online streaming
manner}. We empirically demonstrate that the performance of current approaches
degrades if the input is obtained as a stream of data with the following
restrictions: $(i)$ each instance comes one at a time and can be seen only
once, and $(ii)$ the input data violates the i.i.d assumption, i.e., there can
be a class-based correlation. We propose a novel approach (CIOSL) for the
class-incremental learning in an \emph{online streaming setting} to address
these challenges. The proposed approach leverages implicit and explicit dual
weight regularization and experience replay. The implicit regularization is
leveraged via the knowledge distillation, while the explicit regularization
incorporates a novel approach for parameter regularization by learning the
joint distribution of the buffer replay and the current sample. Also, we
propose an efficient online memory replay and replacement buffer strategy that
significantly boosts the model's performance. Extensive experiments and
ablation on challenging datasets show the efficacy of the proposed method.
