| Summary: | It has been shown that a self-mapping with exactly one removable or jumping discontinuity may have a $ C^1 $ smooth iterate of the second-order. However, some examples show that a self-mapping with exactly one oscillating discontinuity may also have a $ C^1 $ smooth iterate of the second-order, indicating that iteration can turn a self-mapping with exactly one oscillating discontinuity into a $ C^1 $ smooth one. In this paper, we study piecewise $ C^1 $ self-mappings on the open interval $ (0, 1) $ having only one oscillating discontinuity. We give necessary and sufficient conditions for those self-mappings whose second-order iterates are $ C^1 $ smooth.
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