‘Don't’ is a contraction of ‘do not’, and ‘do’ is a verb meaning ‘to perform/execute’. Strictly speaking, then, are these two common constructions ungrammatical?
a) ‘Don't do this/that.’ Since it expands to ‘Do not do this/that’, isn't [1] the second ‘do’ redundant? If so, why does ‘Don't this/that’ sound so wrong whereas the expanded ‘Do not this/that’ merely sounds a bit archaic?
b) ‘Don't continue.’ [2] Since both ‘do’ and ‘continue’ are verbs, shouldn't it be ‘Continue not’? [3]
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[1] ‘Isn't’ seems to follow this remain-grammatical-albeit-archaic-sounding-when-expended rule, as does ‘shouldn't’ in question (b).
[2] Ditto for any other verb in place of ‘continue’.
[3] And if you wanted everyone to think you were odd you could presumably insist on writing ‘continuen't’.