How to check if a file is gzip compressed?

There is a magic number at the beginning of the file. Just read the first two bytes and check if they are equal to 0x1f8b.

Do you prefer false positives, false negatives, or no false results at all (there goes performance down the drain...)?

The RFC 1952: GZIP file format specification version 4.3 states the first 2 bytes (of each member and therefore) of the file are '\x1F' and '\x8B'. Use that for a first check that can result in false positives.