A gift is not deductible (unless given to a qualified charity) by the giver nor is it reportable income by the recipient.
However, the giver may be required to file a gift tax return (separate from your income tax return). "Gift Tax" is somewhat of a misnomer. Even though a gift tax return may be required, very few people ever actually pay federal gift tax. The purpose of the gift tax return is usually only to document a reduction in the allowable estate tax exemption.
See https://turbotax.intuit.com/tax-tools/tax-tips/Tax-Planning-and-Checklists/The-Gift-Tax-Made-Simple/...
For gift tax purposes, the value is $40K. But, for purposes of a future sale, the recipients cost basis is the same as the your cost basis, what you paid for it + any improvements. The $40K appraised value would only be relevant if the recipient sells the house, in the future, at a loss.