A both nondifferentiable (or discontinuous ) and noninvertible one-dimensional map can display a new type of intermittency. The mechanism of it is that a stable fixed point collides with the nondifferentiable of discontinuous point of the map and then disappears. This type of intermittency can happen in the case when the absolute eigenvalue of the linearised map near the fixed point equals any number smaller than 1 before the intermittency. Therefore, it can appear suddenly in any part between a period-doubling cascade, interrupt the cascade and lead to chaos. In the intermittent time plot after disappearence of the stable fixed point the dura-tion of the laminar phase follows a logarithmic dependence on the distance between a control parameter value and its critical number. This new scaling law is independent of the details of the map. We believe that this new type of intermittency should exist in many practical sys-tems.