According to Catholic Church, to what extent is faith a gift of God in terms of Romans 12: 3?
3 For I say, by the grace that is given me, to all that are among you, not to be more wise than it behoveth to be wise, but to be wise unto sobriety, and according as God hath divided to every one the measure of faith.
Faith is one of the three theological virtues that is infused in the believer, by which the intellect is perfected by a supernatural light.
The theological virtues are three: faith, hope, and charity.
Faith is an infused virtue, by which the intellect is perfected by a supernatural light, in virtue of which, under a supernatural movement of the will, it assents firmly to the supernatural truths of Revelation, not on the motive of intrinsic evidence, but on the sole ground of the infallible authority of God revealing. For as man is guided in the attainment of natural happiness by principles of knowledge known by the natural light of reason, so also in the attainment of his supernatural destiny his intellect must be illumined by certain supernatural principles, namely, Divinely revealed truths.
But not only man's intellect must be perfected with regard to his supernatural end, his will also must tend to that end, as a good possible of attainment. Now the virtue, by which the will is so perfected, is the theological virtue of hope. It is commonly defined as a Divinely infused virtue, by which we trust, with an unshaken confidence grounded on the Divine assistance, to attain life everlasting.
But the will must not only tend to God, its ultimate end, it must also be united to Him by a certain conformity. This spiritual union or conformity, by which the soul is united to God, the sovereign Good, is effected by charity. Charity, then, is that theological virtue, by which God, our ultimate end, known by supernatural light, is loved by reason of His own intrinsic goodness or amiability, and our neighbour loved on account of God. It differs from faith, as it regards God not under the aspect of truth but of good. It differs from hope inasmuch as it regards God not as our good precisely (nobis bonum), but as good in Himself (in se bonum). But this love of God as good in Himself does not, as the Quietists maintained, exclude the love of God as He is our good. With regard to the love of our neighbor, it falls within the theological virtue of charity in so far as its motive is the supernatural love of God, and it is thus distinguished from mere natural affection. Of the three theological virtues, charity is the most excellent. Faith and hope, involving as they do a certain imperfection, namely, obscurity of light and absence of possession, will cease with this life, but charity involving no essential defect will last forever. Moreover, while charity excludes all mortal sin, faith and hope are compatible with grievous sin; but as such they are only imperfect virtues; it is only when informed and vivified by charity that their acts are meritorious of eternal life.
Virtue (Catholic Encyclopedia)
Faith is the theological virtue by which we believe in God and believe all that he has said and revealed to us.
1814 Faith is the theological virtue by which we believe in God and believe all that he has said and revealed to us, and that Holy Church proposes for our belief, because he is truth itself. By faith "man freely commits his entire self to God." For this reason the believer seeks to know and do God's will. "The righteous shall live by faith." Living faith "work[s] through charity."
1815 The gift of faith remains in one who has not sinned against it. But "faith apart from works is dead":81 when it is deprived of hope and love, faith does not fully unite the believer to Christ and does not make him a living member of his Body.
1816 The disciple of Christ must not only keep the faith and live on it, but also profess it, confidently bear witness to it, and spread it: "All however must be prepared to confess Christ before men and to follow him along the way of the Cross, amidst the persecutions which the Church never lacks." Service of and witness to the faith are necessary for salvation: "So every one who acknowledges me before men, I also will acknowledge before my Father who is in heaven; but whoever denies me before men, I also will deny before my Father who is in heaven." Catechism of the Catholic Church