If f(x)=x+|x|+cos⁡π2x and g(x) = sin x, where [.] denotes the greatest integer function, then

f(x)=x+|x|+cos⁡9x,g(x)=sin⁡xSince both f (x) and g(x) are continuous everywhere,f (x) + g(x) is also continuous everywhere,f (x) is non-differentiable at x = 0.Hence, f (x) + s(x) is non-differentiable at x = 0. Now,h(x)=f(x)⋅g(x)=(cos⁡9x)(sin⁡x),x<0(2x+cos⁡9x)(sin⁡x),x≥0Clearly, h{x) is continuous at x = 0. Also,h′(x)=cos⁡xcos⁡9x−9sin⁡xsin⁡9x,x<0(2−9sin⁡9x)sin⁡x+cos⁡x(2x+cos⁡9x),x>0h′0−=1,h′0+=1So, f(x) . g(x) is differentiable everywhere.