In practice, some multi-objective optimization problems exhibit expensive calculation characteristic, called expensive multi-objective optimization problems (EMOPs). Most of the existing expensive multi-objective evolutionary algorithms do not consider the discontinuity of Pareto front (PF), their performance is not ideal when solving EMOPs with discontinuous PF. In view of this, this paper proposes a surrogate-assisted multi-objective evolutionary algorithm with discontinuity-inducing variables (VD-SAMOEA). First, a method combining density clustering and random sampling is developed to identify variables that significantly impact the discontinuity of PF. And then, based on the identified discontinuity-inducing variables, a local exploitation strategy with varying search intervals is presented to improve the algorithm’s search performance for the sparse PF segments. Further, a local surrogate model construction method based on geodesic flow kernel multi-source transfer learning is proposed to improve the accuracy of the local surrogate model on the exploited region. Moreover, a convergence-indicator-guided hybrid global search strategy is proposed to balance the diversity and convergence of the population. Finally, experimental results on 20 test functions and the carbon fiber spinning problem indicate that, compared to 11 typical algorithms, the proposed algorithm can obtain high-quality Pareto optimal solutions with a lower computational cost.