Example 5

The real function equal to 0 for $x = 0$, and to $x \sin 1/x$ for $x \neq 0$, is defined and continuous on $\mathbf{R}$, but has neither right nor left derivative at the point $x \neq 0$.* One can give examples of functions which are continuous on an interval and fail to have a derivative at every point of the interval (I, p. 35, exerc. 2 and 3).