“For Christ is the end of the law for righteousness to everyone who believes” (Romans 10:4, ESV).
Some things in your life need to be brought to an end. There is that sin, those patterns and habits, the struggles and problematic situations. All these cause us to be weary and worn out and they need to be finished.
The Holy Spirit has planted a hunger in the heart of every man, woman, boy and girl in the world today – a desire to live righteously and to do things right. Even those who don't know Jesus, people of other religions, all innately have within them a desire to live life well, to love others well, to try their best.
Because of the pitfalls of sin and Satan, many fall far short and become corrupt in their thinking. But deep within us, we all want victory over sin. We want to be cleansed. When we have invited Christ to be Lord and Savior of all that we are and all that we think and all that we desire, then the Holy Spirit puts in each of us a greater desire, a passion, to be holy and pure and righteous before God.
This word righteousness in Romans 10:4 means: to be in right standing with God; functioning in right behavior, right motives, right mind, right emotions. It means taking things that are wrong and out of sync with God and establishing a right alignment to the plans, purposes, will and heart of the Father. The Bible calls this the pursuit of righteousness.
Now, what is this righteousness? Is it just simply doing the right things? I would suggest not, because many people can do the right things with wrong motives. That is called a religious spirit and today many people are indwelt with that spirit. Outwardly they function and do all the right things, but inwardly they are dead men's bones.
The essence of what Paul said in Romans 10:4 is that Christ has already accomplished the purpose for which the law was given and as a result, all who believe in Him are made right (righteous) with God.