One can check if each of $1, 2, 3, \dots, N$ as an element of $A$ to solve in a total of $O(N)$ time time.

Sample code (C++)

Otherwise, one can sort the elements of $A$ in the increasing order and check if $1, 2, \dots, N$ are in this order to solve in a total of $O (N \log N)$ time.

Sample code (C++)

Sample code (Python)