I have been considering what smoothness and continuity relations of the second derivative say about the first derivative of a function and the function itself. So far, I think
a smooth and continuous second derivative means a smooth and continuous first derivative and function
a continuous but not smooth second derivative means a continously and smooth first derivative and function
a second derivative with a finite discontinuity produces a continuous but not smooth first derivative at this point, but the function itself is continuous ad smooth.
Now for the case of an infinite discontinuity in terms second derivative, I am a bit stuck. From considering an example I know: the infinite square potential well in quantum mechanics, the second derivative has an infinite discontinuity, the first derivative has a finite discontinuity and the function itself is continuous but not smooth. I can't seem to motivate this mathematically though; I just am going by what I know the function looks like.
I am not a mathematician so I do not think I would understand mathematically rigorous explanations for these properties! I am just trying to gain some intuition into them. I would appreciate if anyone could confirm/correct my ideas so far, and give some insight into the last case.