  In this paper, we present a family of bivariate copulas by transforming a
given copula function with two increasing functions, named as transformed
copula. One distinctive characteristic of the transformed copula is its
singular component along the main diagonal. Conditions guaranteeing the
transformed function to be a copula function are provided, and several classes
of the transformed copulas are given. The singular component along the main
diagonal of the transformed copula is verified, and the tail dependence
coefficients of the transformed copulas are obtained. Finally, some properties
of the transformed copula are discussed, such as the totally positive of order
2 and the concordance order.
