f(x) is not defined at x = 1.
For x ≥ 1 , f (x) = $\frac{\left|\mathit{x}-1\right|}{\left(\mathit{x}-1\right)}$ = $\frac{\left(\mathit{x}-1\right)}{\left(\mathit{x}-1\right)}$ = 1
For x < 1, f (x) = $\frac{\left|\mathit{x}-1\right|}{\left(\mathit{x}-1\right)}$ = $\frac{-\left(\mathit{x}-1\right)}{\left(\mathit{x}-1\right)}$ = $\frac{\left(1-\mathit{x}\right)}{\left(\mathit{x}-1\right)}$ = - 1
Thus, range of the function is either -1 or 1 at all the points and is undefined at x = 1