Prove that the function f : R → R is not continuous at xo = f(x) = { 2 +1 r40 if x E Q if x 4 Q Explain why we can construct a sequence {xn}1 of rational numbers converging to v2. Have we proved this already?
Prove that the function f : R → R is not continuous at xo = f(x) = { 2 +1 r40 if x E Q if x 4 Q Explain why we can construct a sequence {xn}1 of rational numbers converging to v2. Have we proved this already?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![Prove that the function \( f : \mathbb{R} \rightarrow \mathbb{R} \) is not continuous at \( x_0 = \sqrt{2} \).
\[
f(x) =
\begin{cases}
0 & \text{if } x \in \mathbb{Q} \\
x^2 + 1 & \text{if } x \notin \mathbb{Q}
\end{cases}
\]
Explain why we can construct a sequence \( \{x_n\}_{n=1}^{\infty} \) of rational numbers converging to \( \sqrt{2} \). Have we proved this already?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9128905-2bdf-41e5-8502-b679e3fbc062%2Fc725f257-3181-4f62-a54b-737ae9fc3582%2Ftedk79_processed.png&w=3840&q=75)
Transcribed Image Text:Prove that the function \( f : \mathbb{R} \rightarrow \mathbb{R} \) is not continuous at \( x_0 = \sqrt{2} \).
\[
f(x) =
\begin{cases}
0 & \text{if } x \in \mathbb{Q} \\
x^2 + 1 & \text{if } x \notin \mathbb{Q}
\end{cases}
\]
Explain why we can construct a sequence \( \{x_n\}_{n=1}^{\infty} \) of rational numbers converging to \( \sqrt{2} \). Have we proved this already?
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