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Must can be used in reported speech.

  1. I told him that he must stay home.

However, what happens if we use other forms of the infinitive?

  • What is Jack doing?
  • He must be watching TV.

2)I said that Jack must be watching TV. or I said that Jack must have been watching TV.

user1425
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    I said that Jack must have been watching TV only applies if the original question and reply had been Q: What was* Jack doing?* A: He must have been* watching TV.* But note that this isn't the same as your first example, where someone was obliged to stay home. The equivalent for Jack would normally be He had to* watch TV, because we don't like using must* in past tense constructions. See Is “must” ever grammatical as a past tense verb? (possible duplicate?). – FumbleFingers Oct 16 '20 at 11:27
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    All modals can be used in reported speech. They can also be used in direct speech. They have no relation to this distinction. And paying so much attention to tense, a category that is disappearing and of no real importance in English, is a waste of time. – John Lawler Oct 16 '20 at 15:56
  • What is not a waste of time? – user1425 Oct 16 '20 at 15:56

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