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I have been attempting to do the following question by contradiction. However, I just got stuck at where to use the given continuity condition. It would be really appreciated if you can possibly give a further hint to solving this problem.

Here it goes

Let $f(x):\mathbb{R} \rightarrow \mathbb{Q}$ is a continuous function. Prove that $f$ is a constant.

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Can you find a real number between two given rationals? –  Pedro Tamaroff May 5 '12 at 0:18
    
More to the point, can you find an irrational number between two rationals? –  Robert Israel May 5 '12 at 0:31
    
Yes, it was my initial idea of using Density Theorem. However, I do not see the connection between that with the function. –  Viet Hoang Quoc May 5 '12 at 0:42
    
Look at this –  leo May 5 '12 at 0:49

1 Answer 1

The basic idea behind this problem is the following statement:

Theorem: The continuous image of a connected space is connected.

Now, $\mathbb{R}$ is connected--that is just a fact. Now, $\mathbb{Q}$ is not connected (in fact, it's totally disconnected) since given any irrational number $\xi$ one has that $\mathbb{Q}=[(-\infty,\xi)\cap\mathbb{Q}]\cup[(\xi,\infty)\cap\mathbb{Q}]$ is a disconnection.

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Great solution, however, the question requires the answer to use the definition of the continuity. Thereby, I just do not know where to start. –  Viet Hoang Quoc May 5 '12 at 0:39
    
@VietHoangQuoc Well, which definition of continuity are you using? –  Alex Youcis May 5 '12 at 0:41
    
The formal one as follows For all $ \epsilon >0$, there exists $ \delta >0$ such that for all $ x \in \mathbb{R}$ satisfying $ |x-x_0| < \delta$, one has $ |f(x)-f(x_0)| < \epsilon $. –  Viet Hoang Quoc May 5 '12 at 0:43
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@VietHoangQuoc Ditch that definition here. Use the following definition: A function $f: X \rightarrow Y$ between two topological spaces is continuous if for every open set $U$ in $Y$, the preimage of $U$ under $f$ is open in $X$. –  user38268 May 5 '12 at 0:46
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Here is another similar type of question proving the function is constant using the continuity definition, Let $f: [-1,1] \to \mathbb{R}$ be a function with $f(x)=f(x^2)$ for all $ x \in (-1,1)$. Suppose that $f$ is continuous at 0. Prove that $f$ is a constant. –  Viet Hoang Quoc May 5 '12 at 1:03

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