Verification/Conviction: Just to simply make sure that a solution is correct when there may be some uncertainties.

Explanation: Would be used when someone wants to know why a certain procedure or formula makes sense. Maybe when someone wants to know why an integral can be used to determine an area.

Discovery: This function of proof would be valid for exploring a topic and either proving or disproving.

Systematization: This function could be used to clearly show why something is true and how the steps relate to one anotyher.

Communication: THis function serves as a way of explanation between people. It makes things clearly communicable in a common language.

Intellectual challenge: Could serve as a way to provide entertainment and fun through working proofs. It can make math both fun and educational.