This is easily solved in practice by segmentation. Images are segmented in blocks, and each small block can be well approximated by a low frequency content, but

This is easily solved in practice by segmentation. Images are segmented in blocks, and each small block can be well approximated by a low frequency content, but block boundaries can contain discontinuities. If you look at the wolfram alpha source, that's exactly how those images are constructed -- they are combinations low frequency segments, most likely obtained through fourier analysis. Another trick used in frequency domain compression is they don't impose a hard cut-off of frequencies (truncation); instead they specify weights according to image quality, and lower are assigned to high frequencies, and those weights control the precision of each frequency. All this works because real signals happen to be concentrated in certain frequencies, with a few discontinuities in between.