# Rationals/Irrationals and Continuity

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• February 27th 2010, 03:21 PM
CrazyCat87
Rationals/Irrationals and Continuity
If you define $g: \mathbb{R} \rightarrow \mathbb{R}$ by $g(x) := 2x$ for $x$ rational, and $g(x) := x + 3$ for $x$ irrational, at what points is $g$ continuous?
• February 27th 2010, 10:03 PM
Drexel28
Quote:

Originally Posted by CrazyCat87
If you define $g: \mathbb{R} \rightarrow \mathbb{R}$ by $g(x) := 2x$ for $x$ rational, and $g(x) := x + 3$ for $x$ irrational, at what points is $g$ continuous?

Think about it likes this. If $f$ is continuous and $x_n\to x$ then $f(x_n)\to f(x)$. So, let $x\in\mathbb{R}$ the since both the irrationals and rations are dense in the reals there exists sequences $\{q_n\}_{n\in\mathbb{N}},\{i_n\}_{n\in\mathbb{N}}$ such that $q_n\to x,i_n\to x$. Thus, we must have that $f(x)=\lim\text{ }f(q_n)=\lim\text{ }2q_n=2\lim\text{ }q_n=2x$ and $f(x)=\lim\text{ }f(i_n)=\lim\text{ }\left\{i_n+3\right\}=\lim\text{ }i_n+3=x+3$. In particular, $2x=x+3\implies x=3$. Thus, that is the only point of continuity.