Real saving faith that becomes false faith without salvation is an oxymoron. Either the word of God (through which faith comes) is on the good soil of our heart or it isn't. If it isn't then it wasn't real faith to begin with because it lacked a solid foundation.
Falling away comes from false doctrines, impotent promises never proposed by God to begin with, such as works-based salvation for example. One will eventually realize their sheer inadequacy to attain and hold the level of perfection God requires for very long, if at all.
That makes repentance a change of mind leading to a change of actions. If one sins after repentance, was it true repentance? If repentance is an effort to make ourselves self-righteous before God then why does He call our righteousness a filthy rag? You got it all wrong.
It is God who sanctifies us, disciplines us, and makes us righteous through Christ, the worthy sin offering. This is not about us walking in sinless perfection from the exact moment we come to have faith in Christ. If that were so then conversion from a worldly lifestyle to Child of God would be impossible because new Christians are often worldly. Mature Christians have had more time to be refined by God.
He who began a good work in us will complete it. Just have faith.