This paper deals with continuous tension validation for Cable-Driven Parallel Robots (CDPRs). The proposed method aims at determining whether or not a quasi-static path is feasible regarding cable tension limits. The available wrench set (AWS) is the set of wrenches that can be generated with cable tensions within given minimum and maximum limits. A pose of the robot is considered valid regarding the tensions if and only if the wrench induced by the platform weight is inside the AWS. The hyperplane shifting method gives a geometric representation of the AWS as the intersection of half-spaces. For each facet-defining hyperplane of the AWS, we define a value which is positive when the pose is valid, i.e. when the corresponding wrench lies on the proper side of the hyperplane. Using this value and an upper bound on its time derivative along the path, the half-length of a valid time interval is obtained. Intervals are repeatedly validated for each hyperplane until either the whole path is validated or a non-valid pose is found. The presented method is integrated within the open-source software Humanoid Path Planner (HPP) and implementation results using the configuration of the CDPR CoGiRo are presented.