Could be solved a NP-complete problem in constant time?
Write down the definition of NP-complete. Then take your assumption that P = NP, so in the definition of NP-completeness you can replace "NP" with "P". Figure out which functions exactly are NP-complete. (Hint: There is a simple and obvious solution that is almost but not quite correct. Read the definition carefully). Do these functions contain any constant time solvable problems? 

Each step is actually quite simple, you just have to follow all the definitions very precisely.