This paper presents a stability analysis method for discrete-time Takagi-Sugeno fuzzy dynamic systems based on a piecewise smooth Lyapunov function. It is shown that the stability of the fuzzy dynamic system can be established if a piecewise Lyapunov function can be constructed, and moreover, the function can be obtained by solving a set of linear matrix inequalities that is numerically feasible with commercially available software. It is also demonstrated via numerical examples that the stability result based on the piecewise quadratic Lyapunov functions is less conservative than that based on the common quadratic Lyapunov functions.