The Exact Batting Average

Source

I got this problem from Rustan Leino, who Bertrand Meyer, who had heard it was once given on the Putnam exam.

I solved it and wrote up my solution.

Problem    

At some point during a baseball season, a player has a batting average of less than 80%. Later during the season, his average exceeds 80%. Prove that at some point, his batting average was exactly 80%.

Also, for which numbers other than 80% does this property hold?

Solution     Reveal