Flexible displays offer new interaction techniques such as bend gestures, but little work has been done to support touch input. Touch is the most common input method for handheld devices. We explore the potential of touch input using the thumb of the holding hand on flexible displays. We conducted two studies to evaluate tapping and swiping with thumb input, and compared the interactions between a flexible and a rigid tablet. Our result suggests that participants can perform tapping and swiping using thumb input similarly for both rigidities. This indicates that thumb input can offer the benefits of touch input in flexible displays. Our results also suggest that users prefer holding the display on the side or on the bottom corner over the bottom center. We present design guidelines to use thumb input on flexible devices, and provide three use case scenarios to visualize our design guidelines.