2
$\begingroup$

At which points is the following function continuous?

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

  • 0
    I've done nothing because I have no idea where to start from.2012-01-24
  • 0
    $Q$ is the set of rational number?2012-01-24
  • 0
    When is $5x = x^2-6$?2012-01-24
  • 0
    Where are the alternatives equal?2012-01-24
  • 0
    @Paul: Yes, I think so. it's not mentioned in the question.2012-01-24
  • 0
    @lhf: Only at $2$ and $3$ ?2012-01-24
  • 1
    @Gigili, no, -1 and 6.2012-01-24

1 Answers 1

1

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 $x$.

  • 3
    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