show that there exists an increasing function

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?

Re: show that there exists an increasing function

Quote:

Originally Posted by

**wopashui** hint:[For x

[0,1], define g(x)=

{f(t): 0<=t<=1, t

Q}]

It should say: .

Re: show that there exists an increasing function

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.