No False Peace
Some sentimental people think that God's job is to be nice to everybody, so that they can enjoy His blessings and get His help when they want it, ignoring Him when they are managing fine (Mark 7:6). However, that is no way to relate with the God revealed in the Bible, whose character and nature is seen in Jesus Christ (Hebrews 1:1-4)! God is loving and kind and caring and compassionate (Psalm 116:5); but He is also true and just and righteous (Psalm 145:17) and hates evil (Proverbs 6:16-19). All those who seek His Kingdom will love Him, but others will resist His Kingly demands and hate His authority. They will also hate those who love Him; even hating members of their own family.
Jesus was preparing His disciples for the many waves of persecution that lay ahead for the church. It would be felt most intensely in families. What a paradox, because God is the great Father and He 'invented' the family (Ephesians 3:14-15)! But if people rebel against His love and refuse to be a part of His family they may even reject their own blood relatives who love Him.
It is not that Jesus is a troublemaker: no - He wants to bring His peace to everybody. Until Christ comes into a family, He is unknown ... there is nothing to rebel against, and the whole family muddles on towards hell together. But when the Light of Christ is lit in one person's heart, the others in the family are presented with a challenge they never expected - to see their darkness for what it is (John 3:19-20). Some will refuse and may separate themselves from the believer.
That still happens today. Many followers of Jesus face the same challenges in their family life. What is the solution? We cannot go back to being what we were because by God’s grace we have found the truth which has set us free. However much we may (and we should) seek to explain the gospel to them, we cannot force any of our family to believe in Jesus and obey Him: they must respond to His call for themselves. So, as far as it is possible, we should live peaceably with everyone: and we should not get revenge on those who treat us badly (Romans 12:18-19). However, we can pray for those who persecute us (Luke 6:27-28) and let the light of Christ shine from our lives (Matthew 5:16). And we should never forget that Jesus spoke a special blessing over all those who suffer because they are trusting Jesus. Matthew 5:11-12 assures us of a great reward in heaven. So take courage!