Decision problems: Does a "constant" characteristic function describe a language in P?
Yes, of course they're in&nbsp;P: you can decide them in constant time, which is certainly bounded by a polynomial.

Anybody who tells you that P$\,=\,$NP implies that every problem in&nbsp;P is NP-complete has forgotten about these two trivial problems. Neither of them can be NP-complete, because NP-completeness is defined in terms of many-one reductions, but there cannot be a many-one reduction from a non-trivial probelm to either of the trivial problems. A many-one reduction from $A$ to&nbsp;$B$ must map "yes" instances of&nbsp;$A$ to "yes" instances of&nbsp;$B$, and "no" instances to "no" instances.  But one of the trivial problems has no "yes" instances and the other has no "no" instances, so a problem that has both "yes" instances and "no" instances cannot be reduced to either.