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
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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?
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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