Using continuous controlled liftings of vector fields, we first prove for Bekka's (c)-and hence Whitney (b)-regular stratifications S that near every point of a stratum X with depth Σ (X) = 1 there exists a local C 0,1 foliation. Then we construct a local open book structure near each point of X and use this result to prove the general smooth version of the Whitney fibering conjecture near every point of an arbitrary stratum X of S. As a consequence we improve the Thom-Mather regularity of the local trivialization maps of a proper stratified submersion f : S → M into a manifold. In a famous paper of 1965 H. Whitney proposed a local fibering property around points of a complex analytic variety. More precisely he conjectured that every complex analytic variety V admits a stratification such that a neighbourhood U of each point is fibered by copies of the intersection of U with the stratum M containing the point. He asked also that the fibers be holomorphic manifolds and that their tangent spaces vary continuously as nearby points approach M. Note that if one does not require the continuity of tangent spaces to the fibers then the Thom-Mather isotopy theorem suffices to prove a smooth version of Whitney's conjecture. In 1989 R. Hardt and D. Sullivan gave a proof of a similar conclusion for holomorphic varieties but again without the essential continuity of the tangent spaces to the fibers. From 1993 the first author studied the possibility of obtaining the analogous property in the case of smooth real stratified spaces in his thesis under the direction of the third author who conjectured this property be true for Whitney (b)-regular stratifications.