This question has been the cause of an almost unending debate. The simplest answer is that though Hosea’s ministry was conducted in the north, his message was continued through either disciples or in its written form in the south—possibly both. This is not a problematic view, when we consider Paul’s testimony that all of God’s inspired words are useful for all of God’s people (2 Timothy 3:16), not only the immediate and original audience.
While sceptics insist that a prophet to the north would not have been concerned with Judah, we should have no problem affirming that God always has his whole people in view.1 For example, Hosea promises reunification under a single head (Hosea 1:11), the Davidic king (Hosea 3:5). Hosea’s message was for all of God’s people, from Israel to Judah, Samaria to Jerusalem. The impact of Hosea’s prophecies continued to be felt in Judah long after the tumultuous collapse of Samaria; indeed the very fulfillment of his sad oracles encouraged those who had preserved them to continue to hear in them the living word of God.
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1 The word of the LORD that came to Hosea, the son of Beeri, in the days of Uzziah, Jotham, Ahaz, and Hezekiah, kings of Judah, and in the days of Jeroboam the son of Joash, king of Israel.