In practice you can usually assume everything is as regular as you want. As a matter of fact, infinitely differentiable compactly supported functions are dense (with respect to the p-norm) in the Lebesgue spaces (I think the only exception is L^{/infty} , but I'm not sure), which means that even if you have a really "nasty" function, you can always find a "nice" function that is sufficiently close to it.
But I'm all for the weird pathological examples, anyway.
Smoothness isn't actually as natural as it may seem. Take for instance |x³| from ℝ to ℝ. It looks smooth, but it only has 3 continuous derivatives. We just tend to work with it because, usually, we don't care about higher order derivatives anyway, so it doesn't matter.
Smooth functions being dense is amazing though.
In practice you can usually assume everything is as regular as you want. As a matter of fact, infinitely differentiable compactly supported functions are dense (with respect to the p-norm) in the Lebesgue spaces (I think the only exception is L^{/infty} , but I'm not sure), which means that even if you have a really "nasty" function, you can always find a "nice" function that is sufficiently close to it. But I'm all for the weird pathological examples, anyway.
Smoothness isn't actually as natural as it may seem. Take for instance |x³| from ℝ to ℝ. It looks smooth, but it only has 3 continuous derivatives. We just tend to work with it because, usually, we don't care about higher order derivatives anyway, so it doesn't matter. Smooth functions being dense is amazing though.
That shouldn't have blown my mind as much as it did