Prove that the function f given by `f(x)" "=" "|" "x" "" "1|," "x in R` is not differentiable at `x" "=" "1`

Explanation

`f(x) =| x - 1 | = {{:(x - 1 "," if x ge 1),(-(x - 1)"," if x lt 1):}`
`f(1) = 1 - 1 = 0`
` Lf '(1) = underset (h to 0) (lim) (f(1 - h)-f(1))/(-h)`
`= underset (h to 0) (lim) (-(1- h- 1-0))/(-h)`
` = underset (h to 0) (lim) (h)/(-h)= underset(hto 0)(lim)(-1)= -1`
Rf'(1)= `= underset (h to 0) (lim)(f(1 +h)-f(1))/(h)`
`= underset (h to 0) (lim)((1+ h- 1)-0)/(h)`
` = underset (h to 0) (lim) (h)/(h)= underset(hto 0)(lim)(1)= 1`
`:'` Lf' (1) `ne`Rf' (1)ltbr gt `:.` f(x) is not differentiable at x = 1 . Hence Proved .