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Math Help - Rationals/Irrationals and Continuity

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    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?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by CrazyCat87 View Post
    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.
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