continual righteousness through Christ

Rom 1.17 For in the gospel the righteousness of God is revealed—a righteousness that is by faith from first to last, just as it is written: “The righteous will live by faith.”

What set the Gospel apart from the Law was the fulfillment of Christ, who enabled righteousness to last, from beginning to end. Sin could not disqualify a man whose faith was in Christ, who knew no sin.

The Law contained all kinds of ways of mitigating sin so that a man and a nation could remain blessed by God and in fellowship with Him. But the sin nature remains within man, and the Law could not get past this disqualifying factor.

But the Gospel offered Christ as the ultimate mitigating factor which does overcome this disability. Faith in Christ enables us to pursue his righteousness continually, without disqualification, because his mercy never stops advocating on our behalf, restoring us continually to God.

Romans 10:3-4
For they being ignorant of God's righteousness, and going about to establish their own righteousness, have not submitted themselves unto the righteousness of God. 4For Christ is the end of the law for righteousness to every one that believeth.

10 For since the law has but a shadow of the good things to come instead of the true form of these realities, it can never, by the same sacrifices that are continually offered every year, make perfect those who draw near. 2 Otherwise, would they not have ceased to be offered, since the worshipers, having once been cleansed, would no longer have any consciousness of sins? 3 But in these sacrifices there is a reminder of sins every year. 4 For it is impossible for the blood of bulls and goats to take away sins.
5 Consequently, when Christ came into the world, he said,
“Sacrifices and offerings you have not desired,
but a body have you prepared for me;
6 in burnt offerings and sin offerings
you have taken no pleasure.
7 Then I said, ‘Behold, I have come to do your will, O God,
as it is written of me in the scroll of the book.’”
8 When he said above, “You have neither desired nor taken pleasure in sacrifices and offerings and burnt offerings and sin offerings” (these are offered according to the law), 9 then he added, “Behold, I have come to do your will.” He does away with the first in order to establish the second. 10 And by that will we have been sanctified through the offering of the body of Jesus Christ once for all.
11 And every priest stands daily at his service, offering repeatedly the same sacrifices, which can never take away sins. 12 But when Christ had offered for all time a single sacrifice for sins, he sat down at the right hand of God, 13 waiting from that time until his enemies should be made a footstool for his feet. 14 For by a single offering he has perfected for all time those who are being sanctified.
15 And the Holy Spirit also bears witness to us; for after saying,
16 “This is the covenant that I will make with them
after those days, declares the Lord:
I will put my laws on their hearts,
and write them on their minds,”
17 then he adds,
“I will remember their sins and their lawless deeds no more.”
18 Where there is forgiveness of these, there is no longer any offering for sin.

Hebrews

@OP, Agreed. Heb 12:14

New International Version
Make every effort to live in peace with everyone and to be holy; without holiness no one will see the Lord.

New Living Translation
Work at living in peace with everyone, and work at living a holy life, for those who are not holy will not see the Lord.

English Standard Version
Strive for peace with everyone, and for the holiness without which no one will see the Lord.

This "faith" is truly a continuous faith that starts with our first believing until our final breath. Jesus spoke of this in two important Scriptures. Those being:

John 3:16 and Matthew 24:13

As far as without "disqualification," this requires our obedience to the Commandments of Jesus. As well as, if we do slip from time to time, our repenting and asking His forgiveness just as He taught in the Prayer He gave us to pray........ Matthew 6:9-13