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The investigation of asymptotic properties of some singularly perturbed boundary value problems and the choice of an appropriate method for the construction of solutions or approximations to solutions often prove to be very difficult. An analysis shows that such problems include boundary value problems leading to infinitely large values of the solution at a discontinuity point as the small parameter tends to zero. Typically, there is an initial jump in such problems [1-3]. The most general results in this direction were obtained in [4-7]. However, the asymptotics of the solution of a boundary value problem is constructed in these papers under the a priori assumption that the derivative of the desired solution or the fast variable is infinitely large as the small parameter tends to zero. We naturally encounter the problem of finding a class of boundary value problems for which the initial jump phenomenon occurs. This is the aim of the present paper. [ABSTRACT FROM AUTHOR]