There are seven New Testament references to sayings (or "words") that are referred to as "faithful," four of which use the phrase "faithful saying."
Five of the references to sayings that are faithful are found in Paul's pastoral epistles in the New Testament.
1. This is a faithful saying, and worthy of all acceptation, that Christ Jesus came into the world to save sinners; of whom I am chief. 1 Timothy 1:15
This first faithful saying tells us one of the reasons why Jesus Christ came to this earth to be born (incarnation), to die (death and burial), and to arise from the dead (resurrection): Christ Jesus came into the world to save sinners. The verse also tells us that Christ did not come to save good people. Jesus said, "I am not come to call the righteous, but sinners to repentance" (Matthew 9:13). The Apostle Paul considered himself the "chief of sinners" and he obtained mercy and life everlasting by believing on the Lord Jesus. Do not think that you are too great a sinner, like I did before coming to Christ. If you have not come to the Lord Jesus Christ, will you not come to Him by faith today and prove the reality of His power to save?
2. This is a true saying, If a man desire the office of a bishop, he desireth a good work. 1 Timothy 3:1
The word "true" in this verse is the same Greek word (Strongs, #4103 "pistos") translated "faithful" in the other verses. This verse teaches that it is good to desire a position of leadership in a local church. The next verses (vs. 2 and following) give the qualifications for the man desiring such an office.
3. But refuse profane and old wives' fables, and exercise thyself rather unto godliness. For bodily exercise profiteth little: but godliness is profitable unto all things, having promise of the life that now is, and of that which is to come. This is a faithful saying and worthy of all acceptation.1 Timothy 4:7-9
Although some people put a lot of emphasis on physical exercise, Christian's should be more concerned about the important things of eternity and exercise themselves unto godliness, which is profitable for this present life as well as the life which is to come.
4. It is a faithful saying: For if we be dead with him, we shall also live with him: If we suffer, we shall also reign with him: if we deny him, he also will deny us: If we believe not, yet he abideth faithful: he cannot deny himself. 2 Timothy 2:11-13
Believers are seen by God as identified with Christ in his death, so we also shall go to live with Christ in heaven, to be forever with Him.
5. That being justified by his grace, we should be made heirs according to the hope of eternal life. This is a faithful saying, and these things I will that thou affirm constantly, that they which have believed in God might be careful to maintain good works. These things are good and profitable unto men. But avoid foolish questions, and genealogies, and contentions, and strivings about the law; for they are unprofitable and vain. Titus 3:7-9
Being justified by God's grace, the believer in Christ is made an heir of eternal life. The Apostle Paul, therefore, instructs Titus to continually tell the believers in Christ that they should be diligent about doing good works, not just occasionally, but to seek opportunity for doing them, for doing this is good and profitable unto men. He then mentions some "unprofitable and vain" things that Christian's should avoid and not do.
The final two references are found in the last two chapters of the Book of Revelation and emphasize that the words in that Book are true and faithful.
6. And he that sat upon the throne said, Behold, I make all things new. And he said unto me, Write: for these words are true and faithful. Revelation 21:5
The truth and certainty of a future new heaven and new earth are assured by the word and promise of God, and are ordered to be committed to writing as a permanent record.
7. And he said unto me, These sayings are faithful and true: and the Lord God of the holy prophets sent his angel to shew unto his servants the things which must shortly be done. Revelation 22:6
The description of the new Jerusalem (heaven) in the verses preceding verse 6 are faithful and true.
All the Bible's sayings are true, but the above mentioned are specifically called "faithful."