Can't open encrypted disk image on Time Machine

I have a password-protected encrypted folder in my Macbook Pro. The folder's name is "Enc.dmg," and is kept in the main Applications folder. If I unmount the the dmg folder before a Time Machine backup, it theoretically seems to back it up. But when I later enter Time Machine and go into Applications, and click on the Enc.dmg file, it doesn't open; it doesn't mount the disk image and prompt me to enter the password. Instead, a non-functional information box pops up. The box contains useless information. It just tells me that it's a dmg file.

If my computer ever crashes or is lost/stolen, and I can't access my encrypted file on Time Machine, I'll be in big trouble. Is there any way to actually make a dmg function properly in Time Machine mode?