Answer to Question #239363 in Real Analysis for dakj

Question #239363

Show that the function defined as 𝑓(𝑥) = { 𝑥, 𝑥 ∈ ℚ

𝑥 2 , 𝑥 ∈ ℝ − ℚ is continuous at 1 and discontinuous at 2.


1
Expert's answer
2021-09-20T15:48:37-0400

f(x)={x     if xQx2   if xRQTo show that f is continuous at 1. We must show that limx1f(x)=f(1)Suppose f(x)=x limx1f(x)=1Suppose f(x)=x2 limx1f(x)=1 Since f(x)=1, then f is continuous at 1.f(x)=\begin{cases} x ~~~~~ \text{if } x\in \mathbb{Q}\\ x^2 ~~~ \text{if } x \in \R-\mathbb{Q} \end{cases}\\ \text{To show that } f\text{ is continuous at 1. We must show that } \\ \lim_{x\to 1} f(x)=f(1)\\ \text{Suppose } f(x)=x\\ \ \lim_{x\to 1} f(x)=1\\ \text{Suppose } f(x)=x^2\\ \ \lim_{x\to 1} f(x)=1 \text{ Since } f(x)=1, \text{ then } f \text{ is continuous at } 1.\\\\

b.

Consider the neighborhood of 2(21n,2+1n)nNThen the neighborhood contain an irrational number xnnN. Since irrational number is dense in R.i.e.xn(21n,2+1n)    21n<xn<2+1n    1n<xn2<1n    xn2<1n    xn2 as nNow, f(xn)=xn2  nN as xn is irrationallimnf(xn)=limnxn2=(limnxn)2=22=4f(2)=2Thus, f is discontinuous at 2\text{Consider the neighborhood of 2}\\(2-\frac{1}{n}, 2+\frac{1}{n}) n\in \N\\ \text{Then the neighborhood contain an irrational number } x_n n\in \N. \text{ Since irrational number is dense in } \R.\\ i.e. x_n\in (2-\frac{1}{n}, 2+\frac{1}{n})\\ \implies 2-\frac{1}{n}<x_n<2+\frac{1}{n}\\ \implies -\frac{1}{n}<x_n - 2<\frac{1}{n}\\ \implies |x_n-2|<\frac{1}{n}\\ \implies x_n \to 2 \text{ as } n\to \infty\\ \text{Now, } f(x_n)=x_n^2 ~~\forall n\in \N \text{ as } x_n \text{ is irrational}\\ \therefore \lim_{n\to \infty}f(x_n)= \lim_{n\to \infty} x_n^2=(\lim_{n\to \infty} x_n)^2=2^2=4\neq f(2)=2\\ \text{Thus, } f \text{ is discontinuous at } 2



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