atoi — how to identify the difference between zero and error?

As described by @cnicutar and @ouah, atoi can't distinguish a valid 0 from an invalid string making the strtol family better options.

But Why? and How? First understand that both atoi and strtol only convert the initial set of numbers in a string to numeric values. Any trailing non-numeric characters are simply ignored. strtol can be used to check for invalid strings because in addition to a numeric value, it also returns a pointer to the end of the numeric portion of the string. Thus if this end pointer still refers to the start of the original string, you can tell that there was an error and no characters from the string were converted.

There are a few of other subtleties, as seen in the code example:

long lnum;
int num;
char *end;

errno = 0;

lnum = strtol(in_str, &end, 10);        //10 specifies base-10
if (end == in_str)     //if no characters were converted these pointers are equal
    fprintf(stderr, "ERROR: can't convert string to number\n");

//If sizeof(int) == sizeof(long), we have to explicitly check for overflows
if ((lnum == LONG_MAX || lnum == LONG_MIN) && errno == ERANGE)  
    fprintf(stderr, "ERROR: number out of range for LONG\n");

//Because strtol produces a long, check for overflow
if ( (lnum > INT_MAX) || (lnum < INT_MIN) )
    fprintf(stderr, "ERROR: number out of range for INT\n");

//Finally convert the result to a plain int (if that's what you want)
num = (int) lnum;

Note: If you are sure the input string will be within the valid int range, you can eliminate lnum and simply cast strtol's return directly: num = (int) strtolen(in_str, &end, 10);

You cannot.

atoi cannot detect errors. If the result cannot be represented, atoi invokes undefined behavior. Use strtol instead of atoi.

Secure CERT coding advises to use strtol instead of atoi, read:

INT06-C. Use strtol() or a related function to convert a string token to an integer

That's one of the reasons atoi is sometimes considered unsafe. Use strtol / strtoul instead. And if you have it use strtonum.

The function atoi is more dangerous than you might think. The POSIX standard says:

If the value cannot be represented, the behavior is undefined.

The C99 standard says this also:

7.20.1

The functions atof, atoi, atol, and atoll need not affect the value of the integer expression errno on an error. If the value of the result cannot be represented, the behavior is undefined.