सिद्ध कीजिए कि फलन `f(x)=|x-1|, xin R,x = 1` पर अवकलित नहीं है।

Explanation

`f(x)=|x-1|={{:(x-1",","यदि ",xge1),(-(x-1)",","यदि ",xlt1):}`
`f(1)=1-1=0`
`Lf'(1)=underset(hrarr0)(lim)(f(1-h)-f(0))/(-h)`
`=underset(hrarr0)(lim)(-(1-h-1)-0)/(-h)`
`=underset(hrarr0)(lim)(h)/(-h)=underset(hrarr0)(lim)(-1)=-1`
`Rf'(1)=underset(hrarr0)(lim)(f(1+h)-f(1))/(h)`
`=underset(hrarr0)(lim)(h)/(h)=underset(hrarr0)(lim)=1`
`because" "Lf'(1) ne Rf'(1)`
`therefore" "f(x), x = 1 ` पर अवकलनीय नहीं है ।
यही सिद्ध करना था ।