Consider the function \begin{align*} h(x) = \begin{cases} x &
\text{if } x \in \mathbb{Q} \\ 0 & \text{if } x \in \mathbb{R}
\setminus \mathbb{Q}. \end{cases} \end{align*} Prove it is not
continuous on $\mathbb{R} \setminus \{0\}$.
My working to show that it is not continuous on $\mathbb{R} \setminus \{0\}$ is as follows. However, I am wondering, how can I show it is continuous at $0$? Let $(z_n)$ be any sequence such that $(z_n) \rightarrow 0$, I need to show that $h(z_n) \rightarrow 0$. But how? It is certainly obvious, but I can't seem to formally provide an argument.
First, we show that the above function is not continuous at any real number different from $0$. Let $r \neq 0 \in \mathbb{Q}$, since the set of irrationals is dense in $\mathbb{R}$, then there exists a sequence $(x_n)$ contained in the irrationals such that $\lim_{n \rightarrow \infty} x_n = r$. Note that $h(x_n) = 0$ for all $n \in \mathbb{N}$ so $\lim_{n \rightarrow \infty} h(x_n) = 0$ while $h(r) = r \neq 0$. So $\lim_{n \rightarrow \infty} h(x_n) \neq h(r)$, which shows that $h(x)$ is not continuous at any $r \neq 0 \in \mathbb{Q}$.
Now let $i \neq 0 \in \mathbb{R} \setminus \mathbb{Q}$. Since $\mathbb{Q}$ is dense in $\mathbb{R}$, there exists a sequence $(y_n)$ contained in $\mathbb{Q}$ such that $\lim_{n \rightarrow \infty} y_n = i$. Note that $h(y_n) = y_n$ for all $n \in \mathbb{N}$. Therefore, $\lim_{n \rightarrow \infty} h(y_n) = i \neq 0$ while $h(i) = 0$. So, $\lim_{n \rightarrow \infty} h(y_n) \neq h(i)$, which implies that $h(x)$ is not continuous at any $i \neq 0 \in \mathbb{R} \setminus \mathbb{Q}$.
Best Answer
We will prove that $f$ is continuous at zero.
Let $\epsilon>0$.
We have that $f(0)=0$.
We must find $\delta>0$ such that $\forall x:|x|< \delta \Rightarrow |f(x)|< \epsilon$
Now for $0<\delta< \min\{\epsilon,1\}$ we see that if $x \in (-\delta,\delta) \cap \mathbb{Q}$ then $|f(x)|=|x|< \delta < \epsilon$
If $x \in (-\delta,\delta) \cap \mathbb{R}$ \ $\mathbb{Q}$ then $|f(x)|=0< \epsilon$
In either way $|f(x)|< \epsilon, \forall x \in (-\delta,\delta)$ for the chosen $\delta$, thus $f$ is continuous at $0$.
Now suppose that exists $x_0 \neq 0$, a point of continuity of $f$.
Then from the density of rationals and irrationals in the real line, exist $z_n \in \mathbb{Q}$ and $r_n \in \mathbb{R}$\ $\mathbb{Q}$ such that: $$z_n \rightarrow x_0$$ $$r_n \rightarrow x_0$$ From continuity we have that:
$$f(r_n)=0 \rightarrow f(x_0) \Rightarrow f(x_0)=0$$ $$z_n=f(z_n) \rightarrow f(x_0)$$
Thus $f(x_0)=x_0$ and $f(x_0)=0$ therefore $x_0=0$ a contradiction.
We conclude that $f$ is continuous only at $x_0=0$