In this article, the discontinuities of the theory of heat conduction model with memory-dependent derivatives are emphasized. To analyse the discontinuities, the memory-dependent model is applied to a transient thermo-mechanical process. The fundamental equations of the problem are expressed in the form of a vector matrix differential equation. Applying modal decomposition technique the vector matrix differential equation is solved by an eigenvalue approach in the Laplace transform domain. In order to obtain the solution in the physical domain an approximate method by using asymptotic expansion is applied for short time domain and to analyse the nature of the waves and discontinuity of the solutions. Finally, a suitable Lyapunov function, which will be an important tool to study several qualitative properties, is proposed.