Solution: f (0) = 0; x→0? limf(x)=x→0? lim[x? Cos( 1/x)]=0 because x→0? , cos( 1/x) is a bounded function, x? Is infinitesimal, so x→0? lim[x? cos( 1/x)]= 0 = f(0); ∴f(x) is continuous at point x=0.
And the left derivative at x=0 = f'(0? )= 1;
Right derivative at x=0 =f' (0 )=x→0? limf '(x)=x→0? lim[2xcos( 1/x)-x? sin( 1/x)(- 1/x? )]
=x→0? Lim [2xcos (1/x)+sin (1/x)] does not exist;
Therefore, f(x) is continuous but nondifferentiable at x=0.