# Can a function be differentiable while having a discontinuous derivative?

Recently I came across functions like $x^2\sin(1/x)$ and $x^3\sin(1/x)$ where the derivatives were discontinuous. Can there exist a function whose derivative is not conitnuous, and yet the function is differentiable? If yes, please provide some examples.

• Your second example has a continuous derivative, actually. But the first is an example of an everywhere differentiable function whose derivative is discontinuous. May 4, 2015 at 15:20
• What is your definition of differentiable? May 4, 2015 at 15:22
• It's possible using some piece-wise definitions, at least. See here May 4, 2015 at 15:22
• As an aside, see Volterra's function. May 4, 2015 at 16:19