Given a set of robot joint angles (i.e. 7DoF) $\textbf{q} = [q_1, ... , q_n]$ one can calculate the resulting end-effector pose (denoted as $\textbf{x}_\text{EEF}$), using the foward kinematic map.
Let's consider the vice-versa problem now, calculating the possible joint configurations $\textbf{q}_i$ for a desired end-effector pose $\textbf{x}_\text{EEF,des}$. The inverse kinematics could potentially yield infinitely many solutions or (and here comes what I am interested in) no solution (meaning that the pose is not reachable for the robot).
Is there a mathematical method to distinguish whether a pose in Cartesian space is reachable? (Maybe the rank of the Jacobian) Furthermore can we still find a reachability test in case we do have certain joint angle limitations?