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Definition of Continuous Maps 2

TOPO-WNDPE9

Consider the function:

$$f(x)=\begin{cases} \frac{1}{n }\ \text{if }\ x=\frac{m}{n} \in \mathbb{Q}\ \ \text{(most reduced form)}\\\ 0\ \ \text{if }\ x \notin \mathbb{Q}\end{cases}$$

...on the interval $[0,1]$. Which one of the following statements is TRUE?

A

$f(x)$ is continuous at all rational numbers but discontinuous at all irrational numbers on $[0,1]$.

B

$f(x)$ is not continuous at any rational or irrational number on the interval $[0,1]$.

C

$f(x)$ is continuous at all points on $[0,1]$.

D

$f(x)$ is continuous at all irrational numbers but discontinuous at all rational numbers on $[0,1]$.