$real analysis$

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$real analysis$

check picture please, also its an hour and 30 mins not 2 hours

Real Analysis

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a. Consider the function $f(p,q) = 1$ if $(p,q) \neq 0$ and $f(p,q) =0$ if $(p,q) = 0$.

Take the sequence $a_n :=(\frac{1}{n},\frac{1}{n}) \rightarrow(0,0)$. $\lim f(a_n) = \lim 1 \neq f(\lim a_n) = f(0,0) = 0$

b. Take $A=\mathbb{Q}^2 \subset \mathbb{R}^2$. $\text{Int}(A) = \empty$, $\text{Cl}(A) = \mathbb{R}^2$.

c. Consider the function $f(x) = 1$ if $x \neq 0$ and $f(x) =0$ if $x = 0$. Take $U= (-\frac{1}{2},\frac{1}{2})$. $f^{-1}(U) = \{0\}$, which is not open in $\mathbb{R}$

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$real analysis$

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