Question #107071
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
1
Expert's answer
2020-03-30T06:32:29-0400

We will show that the function is not continuous at any point xR.x \in \mathbb{R}. Consider an arbitrary point xR.x \in \mathbb{R}. It is well known that the set of rational numbers Q\mathbb{Q} is everywhere dense in R.\mathbb{R}. Therefore there exists a sequence {rn}n=1Q\{r_n\}_{n=1}^\infty \subset \mathbb{Q} such that rnxr_n \to x for nn \to \infty. Hence f(rn)=22f(r_n)=2 \to 2 for nn \to \infty. Also it is known that the set of irrational numbers I=RQI=\mathbb{R} \setminus \mathbb{Q} is everywhere dense in R\mathbb{R}. Therefore there exists a sequence {xn}n=1I\{x_n\}_{n=1}^\infty \subset I such that xnxx_n \to x for nn \to \infty. Hence f(xn)=44f(x_n)=4 \to 4 for n.n \to \infty.

So, thus we found two sequences rnxxn,r_n \to x \gets x_n, but f(rn)f(xn)↛0|f(r_n)-f(x_n)| \not \to 0 for nn \to \infty. Therefore ff is not continuous at xR.x \in \mathbb{R}. Now if BRB \subset \mathbb{R} is any subset, then ff is not continuous on BB , because ff is discontinuous at every point.






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