In this work, we consider a discontinuous piecewise monotone map f of the interval [0; 1] onto itself, with N >=2 branches. That is, Lorenz maps for N = 2 and a family of expanding Baker-like maps for N > 2. We determine the necessary and sufficient conditions to have chaos in the whole interval. This condition, which we call of full chaos, is very important in engineering applications, especially those related to grazing bifurcations, as well as in other applied fields.
