You cannot send binary data as email unless you first encode it to the allowed range of ASCII characters. The old-school method is with uuencode, and the new-school method with MIME. Uuencode is still recognized by most modern email readers and it is the easiest to implement in a command line pipe.

Sending the file directly without storing it in a temporary file:

mysqldump -u USERNAME -pPASSWORD DATABASENAME | gzip | uuencode mysql-$(date +\%Y-\%m-\%d-\%H.\%M.\%S).gz | mail -s "Gzipped backup file" user@example.com

This pipe doesn't create a temporary file. Gzip directly outputs the compressed data to uuencode which encodes it in proper ASCII characters. That output is piped to the mail command line program.

The command line may appear broken on your screen due to screen width limitations. In reality, all commands should be on one line.

This is great, thanks so much for sharing this code!