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At which points is the following function continuous?

$\begin{eqnarray*} f(x) = \begin{cases} 5x, &\text{if }x \in\mathbb Q, \\ x^2-6, &\text{if }x \notin\mathbb Q. \end{cases} \end{eqnarray*}$

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    @Gigili, no, -1 and 6.2012-01-24

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Consider any point $x\in\mathbb{R}$ and assume that $f$ is continuous at $x$. You can find two sequences $\{a_n\}\subset\mathbb{Q}$ and $\{b_n\}\subset\mathbb{R}\setminus\mathbb{Q}$ such that $\lim a_n=\lim b_n=x$ (do you know why they exist and/or how to find those?). Now use the Heine property for continuity to say that $5x=\lim 5a_n=\lim f(a_n)=f(\lim a_n)=f(\lim b_n)=\lim f(b_n)=\lim b_n^2-6=x^2-6$ Now you can find $x$.

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    And having found $x$ you must still prove that the function _is_ continuous at $x$, since using just two sequences won't prove it.2012-01-24