Lockable lockset operable by either axial or rotational knob movement

A lockset has an actuator unit, a mounting unit and a latch retractor mechanism. The mounting unit attaches the lockset to a door and supports the actuator unit and latch retractor mechanism. The actuator unit includes inside and outside door knobs and a cam mechanism that move together within the mounting unit. During rotational movement, a first cam of the cam mechanism engages the latch retractor mechanism to retract the latch. The actuator is axially movable along an axis of the actuator unit when one of the knobs is pushed or pulled. During axial movement, a second cam of the cam mechanism engages the latch retractor mechanism to retract the latch. A locking mechanism is selectively actuable to act between the actuator unit and the mounting unit to prevent both rotational and axial movement of the actuator unit relative to the mounting unit, thereby locking the lockset.