This article reports a characteristic crisis observed in a kicked rotor subjected to a piecewise continuous force field. The discontinuity border in the definition range of the twodimensional mapping, which describes the system, oscillates as the discrete time develops so that the forward images of the border form a fat fractal. With a chosen group of parameters the iterations on the fat fractal display chaotic motion, and the transient iterations from the initial values in a certain region of the phase space are attracted to the fat fractal. At a threshold of a control parameter, an elliptic periodic orbit suddenly appears inside the fat strange set, inducing an escaping of the iterations to the elliptic islands around it. The fat chaotic attractor thus suddenly transfers to a fat transient set. The influence of the feature of the crisis on the dependence of the lifetime in the transient set on the control parameter has been analyzed. It is shown that the dependence follows a universal scaling law suggested by Grebogy, Ott and Yorke, and the scaling exponent can be approximated according to the variation rules of the elliptic islands and the measure of the fat fractal as control parameter changes. It is possible to compare the results of the direct numerical computation on the scaling exponent and that obtained with the approximation. They are in good agreement.