Q:let f from R to R be continuous function with mod(f(x)-f(y))>=mod(x-y) for every x,y belongs to R.Is f is one-one?.show that there can't exist 3 points a,b,c belongs to R with a<b<c s.t f(a)<f(c)<f(b).
Take , but , that is, you are assuming f is not 1-1. Then the inequality will not hold and the assumption (not 1-1) is false.
If the function is 1-1, then it will either be increasing, or decreasing. If mod is absolute value, then you will have cases: Case 1 automatically proves it, case 2 can't happen from argument above. In case three, assume there is a c where f(c) is in the middle, and use the inequality to show that this can't happen.