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In this correspondence, an online sequential fuzzy extreme learning machine (OS-fuzzy-ELM) has been developed for function approximation and classification problems. The equivalence of a Takagi-Sugeno-Kang (TSK) fuzzy inference system (FIS) to a generalized single hidden-layer feedforward network is shown first, which is then used to develop the OS-fuzzy-ELM algorithm. This results in a FIS that can handle any bounded nonconstant piecewise continuous membership function. Furthermore, the learning in OS-fuzzy-ELM can be done with the input data coming in a one-by-one mode or a chunk-by-chunk (a block of data) mode with fixed or varying chunk size. In OS-fuzzy-ELM, all the antecedent parameters of membership functions are randomly assigned first, and then, the corresponding consequent parameters are determined analytically. Performance comparisons of OS-fuzzy-ELM with other existing algorithms are presented using real-world benchmark problems in the areas of nonlinear system identification, regression, and classification. The results show that the proposed OS-fuzzy-ELM produces similar or better accuracies with at least an order-of-magnitude reduction in the training time.