Answer to Question #107071 in Real Analysis for Garima Ahlawat

Question

Let a function f : R -> R be defined by f(x) = {2, if x belongs to Q , 4 , if x doesn't belongs to Q} check whether f is continuous on B

Answer 1

We will show that the function is not continuous at any point $x \in \mathbb{R}.$ Consider an arbitrary point $x \in \mathbb{R}.$ It is well known that the set of rational numbers $\mathbb{Q}$ is everywhere dense in $\mathbb{R}.$ Therefore there exists a sequence $\{r_n\}_{n=1}^\infty \subset \mathbb{Q}$ such that $r_n \to x$ for $n \to \infty$. Hence $f(r_n)=2 \to 2$ for $n \to \infty$. Also it is known that the set of irrational numbers $I=\mathbb{R} \setminus \mathbb{Q}$ is everywhere dense in $\mathbb{R}$. Therefore there exists a sequence $\{x_n\}_{n=1}^\infty \subset I$ such that $x_n \to x$ for $n \to \infty$. Hence $f(x_n)=4 \to 4$ for $n \to \infty.$

So, thus we found two sequences $r_n \to x \gets x_n,$ but $|f(r_n)-f(x_n)| \not \to 0$ for $n \to \infty$. Therefore $f$ is not continuous at $x \in \mathbb{R}.$ Now if $B \subset \mathbb{R}$ is any subset, then $f$ is not continuous on $B$ , because $f$ is discontinuous at every point.