The Derivative and Higher Derivatives

Abstract

The derivative of a function, even if it exist everywhere, can be discontinuous. For example, if \(f\left( x \right) = {x^2}\sin \frac{1}{x}\), for x ≠ 0, f(0) = 0, then for x ≠ 0, \(f'\left( x \right) = - \cos \frac{1}{x} + 2x\sin \frac{1}{x}\), which does not tend to any limit as x tends to 0, although f′(0) exists and equals 0, as may be easily checked. Thus, f′(x) exists everywhere, but is discontinuous at x = 0.