Do I have to travel every day with a continuous pass?

No, with a 15 days continuous pass (and any other continuous pass), you don't have to travel every day. However, you have the freedom to travel on each day during the overall validity period of your pass.

Example: if you activate your 15 days continuous pass on July 1, you can travel by train as much as you want from July 1 up to and including July 15.