How to prove that a bijective transformation is NOT continuous

Question

I am having this transformation $f: \mathbb R \to \mathbb R$

$$f(x) = \begin{cases} x & x \in \mathbb R \setminus \mathbb Q \\x+1 & x \in \mathbb Q \end{cases}$$

I've already prooved that this transformation is bijective.

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How can I proove that the transformation is NOT continous in every point $x \in \mathbb R $

Answer 1

The easiest way to prove that would be to find a sequence of numbers $a_1,a_2\dots$ such that

$$\lim_{n\to\infty} f(a_n) \neq f\left(\lim_{n\to\infty} a_n\right)$$

To find such a sequence, try to construct it out of rational numbers, but make the limit irrational...