A *jpeg* image? Considering jpeg compression isn't lossless, and it has serious problems with sharp colour changes (which are very common in black-and-white images), you'd be hard pressed to find a black-and-white image, that's still just black and white after a compression/decompression cycle.

Having said that, one way of doing it (untested):

my @out = `djpeg $file.jpg | ppmhist`;
splice @out, 0, 2;
if (@out == 2                          &&
   ($out[0] =~ /^\s*0+\s+0+\s+0\s/     &&
    $out[1] =~ /^\s*255\s+255\s+255\s/ ||
    $out[0] =~ /^\s*255\s+255\s+255\s/ &&
    $out[1] =~ /^\s*0+\s+0+\s+0\s/)) {
    ... is black-and-white ...
}
else {
    ... has colours that are neither black, nor white ...
}
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