Example of a monotonic, everywhere differentiable function on $\Bbb{R}$ whose derivative is not continuous. (English)

Am. Math. Mon. 120, No. 6, 566-568 (2013).

Summary: We construct an example of a monotonic function which is differentiable every-where, but the derivative is not continuous. This is done using a nonnegative discontinuous integrable function for which every point is a Lebesgue point.