Exercises 120p

17. Let $$ f(x) = \begin{cases} x & \quad \text{if } x \text{ is rational}\\ 0 & \quad \text{if } x \text{ is irrational}\\ \end{cases} $$

a. Show that $f$ is countinuous at $x=0$.

Let $\forall \varepsilon,\quad\delta=\varepsilon$

If $x\in \mathbb{Q}$, then $\forall x, \;\; 0<|x-0|<\delta \Rightarrow |f(x)-0|=|x|<\varepsilon$

If $x\in \mathbb{R-Q}$, then $\forall x, \;\; 0<|x-0|<\delta \Rightarrow |f(x)-0|=|0-0|<\varepsilon$

$\blacksquare$

b. Show that $f$ is discountinuous at every nonzero value $x$.

1) Suppose $c\in\mathbb{Q}$ and $\displaystyle{|c|>\varepsilon>\frac{|c|}{2}}$

Every open interval $c \in I$ cotains irrational value $x$ such that $|f(x)-c|=|0-c|=|c|>\varepsilon$

2) Suppose $c\in\mathbb{R-Q}$ and  $\displaystyle{\varepsilon<\frac{|c|}{2}}$

Let $\displaystyle{0<|x-c|<\frac{|c|}{2}}$ and $x\in\mathbb{Q}$, then $$\frac{|c|}{2}<|x|<\frac{3|c|}{2}$$

$\therefore\;\;|f(x)-0|=|x|>\varepsilon$

By 1) 2) $f$ is discontinuous at every nonzero value $x$.

$\blacksquare$

18은 시간이 없어서 손글씨로 올린다.