The paper is concerned with properties of stability for the (piecewise affine) tent map of the interval. The authors consider notions alternative to the topological stability. The first of them is orbit shift stability, which means that for \(C^0\)-small perturbations of a map from the tent family, the conjugacy between the shifts on the corresponding inverse limits, viewed as subsets of the cube \([0,1]^{\mathbb{Z}}\) with a natural metric, remains close to the identity. This is shown not to hold for perturbations within the tent map family. The authors then propose a new concept which they call continuum-wise topological stability (\(cw\)-topological stability). It is conceptually similar to orbit shift stability, but uses set-valued functions, and already holds in the tent family.