Can someone solve this question?
Let $f$ be continuous on $\mathbb{R}$. Let $c$ be in real number with $f(x)=c$ for all $x$ in $\mathbb{Q}$. Show that $f(x)=c$ for all $x$ in $\mathbb{R}$.
thank you
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Julien
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leena adam
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What is the definition of continuity? How is the density of $\mathbb{Q}$ in $\mathbb{R}$ useful? – Sammy Black Apr 11 '13 at 19:34
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Related. – Cameron Buie Apr 11 '13 at 19:54
4 Answers
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You could look at it with sequences, but there is actually a cleaner (but more abstract) way to look at it.
Since $f$ is continuous and $\{c\}$ is closed, $f^{-1}(c)$ is a closed set. But that closed set contains $\Bbb Q$. Therefore... (what can you say about the closure of a dense subset of $\Bbb R$?)
rschwieb
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A subset $D$ of a set $X$ is said to be dense in $X$ if the closure of $D$ is equal to $X$. – rschwieb Apr 11 '13 at 19:53
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Hint: use limits, the density of $\Bbb Q$, and the fact $f$ is continuous.
Clayton
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how we can use limits, the density of Q, and the fact f is continuous. – leena adam Apr 11 '13 at 19:45
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Let $x\in \mathbb R$. Recall a very basic fact: If $f$ is continuous, and $p_n\to x$ then $f(p_n)\to f(x)$.
Now, can you explain why you can find a sequence of rational numbers $(p_n)$ such that $p_n \to x$?
With such a sequence, express $f(x)$ as $\lim_{n\to \infty} f(p_n) $. What do you know about $f(p_n)$? What does that tell you about the limit you just wrote? So, what is the value $f(x)$?
Ittay Weiss
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For any $x\in\Bbb R$, you can find $\left(x_n\right)_{n\in \Bbb N} \in \Bbb Q ^\Bbb N$ so that $x_n \underset{n\to+\infty}{\longrightarrow}x$
The fact that $f$ is continuous tells you that $f$ and $\lim\limits_{n\to+\infty}$ commute, i.e. $f\left(\lim\limits_{n\to+\infty}x_n\right)=\lim\limits_{n\to+\infty}f\left(x_n\right)$
So you have: $f(x)=f\left(\lim\limits_{n\to+\infty}x_n\right)=\lim\limits_{n\to+\infty}f\left(x_n\right)=\lim\limits_{n\to+\infty}c=c$
xavierm02
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