Skip to main content
Mathematics

Return to Question

edited tags; edited title
Link
user21820
user21820
  • 59.2k
  • 9
  • 101
  • 264

Are there known examples of a function that is Everywhere continuous and differentiable at all points in the real number line, but$f : \mathbb{R} → \mathbb{R}$ that is not smooth?

Became Hot Network Question
Source Link
jmarvin_
jmarvin_
  • 23
  • 7

Are there known examples of a function that is continuous and differentiable at all points in the real number line, but not smooth?

I can't seem to find any counterexamples to the statement "all functions that are continuous and differentiable at every point of the reals are smooth," nor can I find anyone asserting or proving this statement. Are there known functions that are continuous and differentiable at every point (with no holes / discontinuities / bounded domain) but are not smooth, that is, after some number of derivatives the derivative function is no longer fully differentiable?