In many two-phase flows and moving interface problems, the flux and the solution of the governing PDE often have finite jumps across the interface. Such jumps often result from source distributions along the interface. One famous application is Peskin's immersed boundary model/method which has been widely used. Using a level set function to represent the interface, we have proposed a new method that can transfer an interface with discontinuity in the flux and/or in the solution to an interface problem with a smooth solution if the coefficient PDE is continuous. The new formulation is based on extensions of the jumps along the normal line of the interface. If the coefficient of the PDE is also discontinuous, then the transformation leads to a new interface problem with homogeneous, also called natural, jump conditions. This will greatly simplify the immersed interface method because no surface derivatives of the jump conditions is needed anymore. Theoretical and numerical analysis including implementation details and numerical example will also be presented.