Jesus said, "Repentance and forgiveness of sins should be preached to all nations" (Luke 24:47).
That shows us that repentance is at the heart of the gospel (God’s message), and forgiveness is not granted without repentance. That's why "There is joy in heaven over one sinner who repents" (Luke 15:10).
In short, "repentance leads to life" (Acts 11:18). Without repentance there can be no life.
A sinner is rebuked in order to produce repentance. Where repentance is evident, there is no need for rebuke. That's why we wish to see repentance.
When the wizard Simon insulted God, Peter rebuked him, and told him that he needed to "repent of this your wickedness and pray" for forgiveness (Acts 8:22).
In the next verse, Peter says to Simon, "For I see that you are poisoned by bitterness and in the bondage of iniquity" (Acts 8:23). When we see such unrepented sin in people, we likewise rebuke them, hoping that as a result of such discipline we will see repentance leading to life.
Of course, repentance itself is invisible. It is a change of heart and we cannot see into people’s hearts (1Corinthians 2:11). However there are "fruits worthy of repentance" (Matthew 3:8) which we can see.
You can't see electricity in a cell, but when electricity flows in a device, it produces things we can see such as light, heat, sound, and motion. Likewise repentance will show visible evidence. We will now consider three such evidences.
People who are repentant will show sorrow because "Godly sorrow produces repentance leading to salvation" (2Corinthians 7:10).
Whilst this sorrow is not a fruit of repentance but rather leads to it, we still may regard it as evidence of repentance. If people do not appear to be sorry, if they show no remorse, we may be sure they are not repentant.
Of course there are two kinds of sorrow: godly sorrow and godless sorrow, but it's not hard to see the difference. Judas felt remorse after betraying Jesus. This was godless sorrow because it led to his suicide rather than to repentance before God (Matthew 27:3-5).
A person who sins but who says, “I have not sinned”, can hardly be considered repentant. Is that not so? Therefore we expect to see confession of sin as evidence of repentance.
Confession of sin shows repentance, but excusing of sin shows a lack of repentance. One must be willing and ready to say before God, “I have sinned’.
John says, "If we confess our sins, God is faithful and just to forgive us our sins and to cleanse us from all unrighteousness. If we say that we have not sinned, we make God a liar and his word is not in us" (1John 1:9-10).
So if people do not confess to God that they have sinned, God will not forgive and cleanse them. And if we do not hear them say, “I have sinned” we have no reason to believe they have repented.
David confessed his sin to God (Psalm 53). The prodigal son confessed his sin (Luke 15:21). So did King Saul (1Samuel 15:24), So did Achan (Joshua 7:20). In these cases, the confession is evidence of repentance.
Of course confession of sin alone is not evidence of repentance. Judas the betrayer, when he felt remorse, said, “I have sinned’ (Matthew 27:4). But he did not, apparently, truly repent. Other fruit worthy of repentance were lacking.
It is not hard to imagine what Judas would have done, had he truly repented, to show that he had made himself right with God. Mainly, he would have kept his apostleship and devoted himself to it. Instead of committing suicide he might have become a martyr for Christ.
Paul, in describing the message he preached, said, "I declared... that people should repent, turn to God, and do works befitting repentance" (Acts 26:20).
John the Baptizer had taught the same: "Bear fruits worthy of repentance" (Matthew 3:8).
Zacchaeus showed his repentance by the manner in which he tried to make amends for his sins (Luke 19:8).
When disciples of Christ do not live so as to show signs of their repentance, they "put Christ to an open shame". The true disciple of Christ will demonstrate repentance with "things that accompany salvation". (Hebrews 6:4-12).