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Let $f:[1,10]\to \Bbb{Q}$ be a continuous function and $f(1)=10,$then $f(10)=?$

$(A)\frac{1}{10}\hspace{1 cm}(B)10\hspace{1 cm}(C)1\hspace{1 cm}(D)$cant be obtained

I could not solve this question.I thought over it for many minutes,here $\Bbb{Q}$ is a set of rational numbers.If $f(1)$ is known,can we find $f(10)$?I suspect,Calculus has something to do in the solution.Can someone please help me in this question?

Brahmagupta
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1 Answers1

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Suppose $f(10) = a \neq 10$ where $a$ is a rational. Then by the IVT, every real number between $a$ and $10$ is attained by the function in the interval $[1, 10]$. (Also, if we are strict since IVT usually requires real functions, we are using a generalization of IVT: continuous functions preserve connectedness.)

Now between every 2 distinct rationals, there is an irrational. So the function must have an irrational number in its image. But the codomain of the function is the rational numbers, so this is a contradiction. Hence $f(10) = 10$.

Credit to the comment above.

Ilham
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    Following a discussion on the chat, this generalization of the IVT is invoked (or at least the reason why the regular IVT can be used needs arguing) because of the phrasing of the question: $f$ is a continuous function from $[a,b]$ to $\mathbb{Q}$, which is not exactly the same as a continuous function from $[a,b]$ to $\mathbb{R}$ such that $f([a,b])\subseteq\mathbb{Q}$ . (There are chances the latter is what the original question intended.) – Clement C. Aug 09 '15 at 07:32