Is there a function f : R → R that is continuously differentiable everywhere but twice differentiable nowhere?

Question

Let Dⁿ denote "n times differentiable" and let Cⁿ denote "n times continuously differentiable".

I'm interested in learning about a counterexample I do not recall ever seeing: a C¹ function

   f : ℝ → ℝ

that is D² nowhere.

And what about a function g : ℝ → ℝ that is D² everywhere but C² nowhere?