You should continue to do it tomorrow.
The sentence above has a future interpretation; "doing" is supposed to be continued in the future.
My question is: what grammatical tense is used in that sentence?
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I think it is preterite (past tense).
By derivation, "should" is the past tense of "shall", but this is mostly unapparent to modern speakers. "should" has developed a life of its own, and is not really perceived as having a "tense" in the normal sense. When used to express obligation, as in your sentence, it is used with a present/future sense, with past obligation being expressed as "should have + pp".
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By derivation, "should" is the past tense of "shall", but this is mostly unapparent to modern speakers. "should" has developed a life of its own, and is not really perceived as having a "tense" in the normal sense. When used to express obligation, as in your sentence, it is used with a present/future sense, with past obligation being expressed as "should have + pp".