To count how many times the digit 4 appears, we need to look at both the tens place and the ones place in the numbers from 1 to 100.
1. In the ones place, 4 appears in 4, 14, 24, 34, 44, 54, 64, 74, 84, and 94. This gives us 10 occurrences of 4.
2. In the tens place, 4 appears in 40 to 49. This gives us 10 more occurrences of 4.
Thus, the total number of times 4 is written is \( 10 + 10 = 19 \).
Therefore, the correct answer is (A) 19.