Make sure there is a memory card inserted in your device and it is compatible with your device. The pictures are not saving on your device, because there isn't enough storage on your device to save them. Try deleting some pictures to create more room.
First, make sure the battery pack is inside the camera. Then make sure the battery is charged. Afterwards, try plugging the camera into an outlet and see if it powers on. If there camera is still not turning on, visit the Troubleshooting Page for the Sony Cyber-Shot DSC-TX55.
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