# Analysis, Proof about, f being continuous, bijective

 P: 4 Let f: R$$\rightarrow$$R be a non-decreasing function. Suppose that f maps Q to Q and f: Q$$\rightarrow$$Q bijection. Prove that f: R$$\rightarrow$$R is continuous, one to one and onto. Hello everyone, I have been staring at this statement for a while now and I just don't understand it, hence I can't even begin to prove it. Can someone explain to me in different words what am I being asked to do. Is f the same same function and Q$$\rightarrow$$Q is somehow "inside" R$$\rightarrow$$R. I don't understand how f could be the same if R and Q don't have the same cardinality, or in other words-I am lost. Any type of help is greatly appreciated.
 P: 610 f maps Q to Q means that if q is in Q, f(q) is also in Q. I am assuming (unless otherwise noted) that Q stands for the rationals.
 P: 4 Yes, Q=rationals, R=reals; but how and why is mapping of Q to Q relevant here, since I am later asked about map of function R to R?
 P: 610 Analysis, Proof about, f being continuous, bijective It is relevant because it is a critical hypothesis for the problem. You are also given that f is non decreasing and that f restricted to Q is a bijection. Think about what the map f restricted to Q looks like. Where does f(q) go to? Now, where does f map the irrational points to?
 P: 4 Since I=irrationals are uncountable and so are reals I guess we could create a bijection between the two (not sure if I am correct here), but I suppose I would be mapped to R.
 P: 93 Ah, I deleted my previous post, what I said was incorrect!. But... let me try to be more helpful: try this outline of a proof (if you like): 1) show that the restriction of f to Q is continuous (treat Q as a metric space on its own and apply the epsilon delta definition). 2) show that there exists a unique extension of f:Q->Q to f:R->R such that f is continuous on R (proving this is fairly easy via contradiction). 3) show that f being non-decreasing on Q and continuous on Q implies it is this candidate function (step 2 isn't entirely necessary, but it should help provide a lot more intuition to make this last leap). I'm quite drowsy, so if something sounds confusing tell me.
 P: 4 I'm sorry, but I'm not familiar with the term "unique extension" what do you mean by that?
P: 93
 Quote by Robert IL I'm sorry, but I'm not familiar with the term "unique extension" what do you mean by that?
Oh no prob. An extension of a function f:Q->Q to a function g:R->R is g such that for every x in Q, f(x) = g(x).

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