The band gap structure of two-dimensional function photonic crystals is investigated with plane wave expansion method, and the refractive index of dielectric cylinder is space coordinate function. Band gap structure of TE and TM waves is analyzed numerically, and which is compared with two-dimensional conventional photonic crystals (the medium columns refractive index are constants). Influences of medium column radius ra, function coefficient k and parameter b on band gap structure of two-dimensional function photonic crystals are investigated. Results show that the band gap adjustment of two-dimensional function photonic crystals can be realize by changing the values of ra, k and b. There are absolute band gaps and semi-Dirac points in two-dimensional function photonic crystals. Results provide a new theoretical basis for the design of optics devices.