Is the change in density continuous or discrete?
I wrote some detailed answers at: https://physics.stackexchange.com/a/139436/59023 and https://physics.stackexchange.com/a/136596/59023.
The density gradient is considered discontinuous because the change occurs on spatial scales comparable to and/or smaller than the mean free path of the particles. It is, of course, not a true discontinuity but ~few micrometers under standard conditions (e.g., STP).
Does aircraft shock wave bend enough light to be seen by naked eyes?
Yep, as you can see in the below image of an F-18 from the Blue Angels, breaking the speed of sound results in a sharp density gradient.
You can see above the tail rudder the places where the index of refraction has changed due to the strong density gradient caused by a density wave driven by the change in the cross-sectional area of the jet (here causing a weak shock). The condensation cone only occurs as the piston transitions from subsonic to supersonic, i.e., during shock initiation due to the rapid change in pressure.
Fun Side Note:
I am fairly confident that the picture was taken in the San Francisco bay area which is interesting since it is illegal to break the sound barrier over populated areas (I think "over land" is the legal phrase used but someone correct me if that is incorrect). The resulting sonic boom would have been rather loud, especially for anyone on the boats in the background.
I do not recall from where I found this image, as it was several years ago (see 2nd update below).
2nd Fun Side Note:
I looked up the sound intensity levels for jet engines while looking for this image (and others) because I recall from the few airshows I attended, military jets are much louder than commercial jets.
One can find a lot of resources on the subject, but one of my favorites is an old hearing protection presentation to the FAA (see the PDF at https://www.faa.gov/about/office_org/headquarters_offices/avs/offices/aam/cami/library/online_libraries/aerospace_medicine/sd/media/McKinley.pdf). At only 30 ft, the F-22 generates upwards of 150 dBA. For reference, pain starts between ~120-130 dBA and the eardrum ruptures between ~140-160 dBA. These jets are so loud that carrier flight deck crews wear both ear plugs and ear head phones and they are still at risk of hearing loss.
The new F-35 is apparently upwards of four times louder (i.e., ~21 dBA louder) than the F-16, which is louder than the F-18.
A colleague of mine informed me that the above photo was taken near San Diego, not San Francisco.
I will correct my first update in that the above photo was taken by Bernard Zee in the San Francisco bay.