✔ 最佳答案
As the other two answers have said, discontinuity of this function follows from the nonexistence of a value for the function at x=1. Of Madhukar's 3 criteria, you appear to be wanting to prove discontinuity solely on the basis of #2.
Another way of stating this is, "There is no possible value for f(1) which would result in continuity at x=1." That can be shown by noting that
|x-1|/(x-1) =
_ { 1, x>1
_ { undefined, x=1
_ {-1, x<1
Then:
lim[x→1+] [|x-1|/(x-1)]*(x+1) = x + 1 = 2
lim[x→1-] [|x-1|/(x-1)]*(x+1) = -(x + 1) = -2
Because the directional limits at x=1 are unequal, the limit doesn't exist, and no possible value for f(1) would make f continuous there.