Let D denote the set of rationals in [0,1] and suppose that f: D--> R is increasing. show that there is an increasing function g:[0,1]-->R such that g(x)=f(x) whenever x is rational.
hint:[For x [0,1], define g(x)= {f(t): 0<=t<=1, t Q}]
according to the hint, we need to show that f is equal to its least upper bound, but f is increasing, so is f(1) the least upper bound?
From what I see, you are supposed to prove that your function has the properties stated. So, check. Is increasing?
Let be chosen arbitrarily, and assume that . Then, is it true that ? Why or why not.
Next, does whenever is rational? Why or why not.