In this paper, the stability and failure tolerant control are investigated for discrete-time time-invariant linear systems with occasional controller failure due to physical or purposeful reason. By introducing the conceptions of minimal controller reliable time and maximal controller failure time, a novel time-varying piecewise quadratic Lyapunov function (TPQLF) approach is proposed to analyse the stability property of closed-loop system in presence of controller failure. The presented TPQLF method combines both merits of previous Lyapunov function based results such as common Lyapunov function and piecewise Lyapunov function approaches. Furthermore, based on TPQLF, the feasible region described by pairs of minimal controller reliable time and maximal controller failure time can be easily computed by an iterative algorithm. Then, on the basis of the analysis results, time-varying state feedback gains are formulated to construct failure tolerant controller. It is demonstrated that the time-varying structure of controller provides more failure tolerance capacity than time-invariant controller. Several numerical examples are given to illustrate our theoretic findings in this paper.