5. If a natural number $n$ allows the vertical addition
$$
n+(n+1)+(n+2)
$$

to not produce any carry, then $n$ is called a "continuous number". For example, 12 is a continuous number, because $12+13+14$ does not produce a carry; but 13 is not a continuous number. Then, the number of continuous numbers less than 1000 is ( ).
(A) 27
(B) 47
(C) 48
(D) 60
5.C.

There are $0,1,2$, a total of 3 single-digit numbers;
There are $\overline{a 0}, \overline{a 1}, \overline{a 2}, a=1,2,3$, a total of $3 \times 3=9$ two-digit numbers;
There are $\overline{a b 0}, \overline{a b 1}, \overline{a b 2}, a=1,2,3, b=0,1,2,3$, a total of $3 \times 4 \times 3=36$ three-digit numbers.
Therefore, there are $3+9+36=48$ continuous numbers less than 1000.