In this paper, we study smoothing approximations for some piecewise smooth functions.We first present two approaches for one-dimensional case: a global approach is to construct smoothing approximations over the whole domain and a local approach is to construct smoothing approximations within appropriate neighborhoods of the nonsmooth points.We obtain some error estimate results for both approaches and discuss whether the smoothing approximations can inherit the convexity of the original
functions. Furthermore, we extend the global approach to some multiple dimensional cases.