This is common for homeowners who purchase a property in mid-late December. If the seller wants a couple of weeks to move out, a buyer can agreed to a sell-leaseback. The new owner will own the property January 1st, but will not occupy the property January 1st (the prior owner is leasing it until the end of January in this example). Because the new owner cannot prove they occupied the property on January 1st, they will be denied the homestead exemption.
A possible way to avoid this
If they had waited to close on the property until January 2nd, the prior owner would have still qualified for a homestead exemption on the property. The homestead exemption would have then carried over for the new owner. Homestead exemptions only care about 1 thing: Did the same person both own and occupy the property on January 1st. So waiting until January 2nd to close on a property is a completely legitimate and legal technique to ensure a homestead exemption stays on the property.