I have dealt with the nature of faith elsewhere and shown that it is the persuasion that a proposition is true (i.e., intellectual assent to a proposition). Saving faith is the persuasion that the saving proposition of the Gospel is true. In this short article, I present my opinion on the function of saving faith in justification.
From God's point of view, saving faith in the elect is His application in time of the justification already wrought both in His purpose before the creation of the world and in time at the crucifixion of Christ. This application, while not required to accomplish their justification, is required by God to achieve His purpose in making a distinction between the elect and non-elect with the intent that proclamation and belief of the Gospel (i.e., God's righteousness in Christ in the vindication of His divine attributes - theodicy) should be the public declarative justification of the former and the public declarative condemnation of the latter, all to praise and wonder of His glory and power.
From the elect's point of view, saving faith is a gift from God that fulfills the condition (although I presume not strictly a required one, such as in the exceptional salvation of elect children who die in infancy and elect mentally impaired people) that enables them to recognize the justification achieved for them at the crucifixion of Christ (and for those blessed to see it, in the purpose of God before the creation of the world), as well as the benefits procured for them by Christ in justification, including forgiveness of sins, imputed righteousness, and peace with God.