Understanding of the continuity and the microscopic structure of as-grown graphene on Cu foils through the chemical vapor deposition (CVD) method is of fundamental significance for optimizing the growth parameters toward high-quality graphene. Because of the corrugated nature of the Cu foil surface, few experimental efforts on this issue have been made so far. We present here a high-resolution scanning tunneling microscopy (STM) study of CVD graphene directly on Cu foils. Our work indicates that graphene can be grown with a perfect continuity extending over both crystalline and noncrystalline regions, highly suggestive of weak graphene-substrate interactions. Due to thermal expansion mismatch, defect-like wrinkles and ripples tend to evolve either along the boundaries of crystalline terraces or on noncrystalline areas for strain relief. Furthermore, the strain effect arising from the conforming of perfect two-dimensional graphene to the highly corrugated surface of Cu foils is found to induce local bonding configuration change of carbon from sp(2) to sp(3), evidenced by the formation of "three-for-six" lattices.