Aparentemente, o infinito e o NaN não fazem parte da especificação JSON, portanto, este código PHP:

$numbers = array();
$numbers ['positive_infinity'] = +INF;
$numbers ['negative_infinity'] = -INF;
$numbers ['not_a_number'] = NAN;
$array_print = print_r ($numbers, true);
$array_json = json_encode ($numbers);
echo "\nprint_r(): $array_print";
echo "\njson_encode(): $array_json";

Produz isso:

PHP Warning:  json_encode(): double INF does not conform to the JSON spec, encoded as 0 in /home/septi/test.php on line 8
PHP Warning:  json_encode(): double -INF does not conform to the JSON spec, encoded as 0 in /home/septi/test.php on line 8
PHP Warning:  json_encode(): double NAN does not conform to the JSON spec, encoded as 0 in /home/septi/test.php on line 8

print_r(): Array
(
    [positive_infinity] => INF
    [negative_infinity] => -INF
    [not_a_number] => NAN
)

json_encode(): {"positive_infinity":0,"negative_infinity":0,"not_a_number":0}

Existe alguma maneira de codificar corretamente esses números sem escrever meu própriojson_encode() função? Talvez alguma solução?