Explain how a digital signature is employed for providing a proof of authenticity (not integrity!) of a file.
A digital signature is generated with a private key and can be validated by anybody who has the public key. So, the digital signature of an object (e.g. a file) ensures that the one who generated the signature (the signer) is the one having the private key. This is called "signer authenticity". As the signature is tied to the content (i.e. a different object would have a different signature), it also verifies that the object (e.g. the file) has not been modified in its way to the receiver.