In astrology is December 23 considered a cusp?

A cusp is the separation between two signs. A planet on a cusp is still in one sign, but may carry over traits from the previous sign. December 23 is the cusp of Sagittarius and Capricorn.
The Date December 23rd isn't considered to be a cusp. A cusp is between 29 degrees 59 minutes and 59 second of one zodiac sign and 0 degrees of the zodiac sign immediately after it; a cusp between Aquarius and Pisces for example.
People born December 23rd are Capricorn, on the cusp (please note, no r) of Sagittarius.