Based on the idea of the extended finite element method (XFEM), a new enriched displacement function, the exponent discontinuous function is presented to simulate the discontinuous character resulted from discontinuity such as crack, interface or joint. Simultaneously, for element containing the crack tip, the triangular basis function is introduced to embody singularity of the discontinuity tip. Referencing to examples of a cantilever deep beam and a rectangular plate, adopting the enriched exponent discontinuous function and enriched step function separately, their calculated results indicate that the results obtained by enriched exponent discontinuous function are more accurate for mode I crack, but as good as by enriched step function for mode II crack. These examples show the effectiveness and superiority of the improved XFEM.