I had studied a lot of Newtonian mechanics and other concepts such as SHMs, Waves and even Thermodynamics but one thing i always noticed was things were continuous and differentiable. All the functions I calculated as answers could be integrated or differentiated to find other parameters however once i started studying Electrostatics and other related concepts i noticed many discontinuous graphs as well. For instance the electric field versus distance graph for a charged spherical shell is discontinuous as you can see in the graph below
Since i had been dealing with only continuous stuff without any breaks it seemed to paint a very smooth picture of the observable nature around me as you don't suddenly see abrupt breaks in the velocity of a particle where it goes from like 0 to 10 without any time lag however these notions tend to start breaking when I dealt with electrostatics.
I know the phrasing of this question might seem a bit weird or philosophical even but my simple question is what does it mean for nature to be discontinuous? I know there a phenomenon which explain why they happen but do these abrupt breaks and non differentiable points in curves reflect some intrinsic properties of the system we are dealing with?
Also in my very short duration of studying actual physics I saw many examples so i reckon there must be much more of these in higher studies. Hence I think they might reflect some fundamental properties of nature or the system but is there any explanation or good reasoning for it being so? Also if it is so prevalent in such systems why is it so that we don't see any of such cases in our day to day life in our immediate surroundings?