Is DNA synthesized continually during the cell cycle

No

DNA is synthesized in the middle part of Interphase, which is usually denoted S in the series G1 - S - G2 "There are 3 stages of interphase: G1 in which the cell grows and functions normally. Once it grows too large it enters the synthesis (S) phase. In S phase the cell makes exact copies of its DNA. It is now ready for G2 phase where it makes sure all of its chromosomes and organelles are ready for mitosis."