Is $f$ and $g$ is Riemann-integrable?

Question

let $f : [0,1] \rightarrow \mathbb{R}$ and $g:[0,1] \rightarrow \mathbb{R}$ be two function define by

$$f(x) = \begin{cases} \frac {1}{n},\text{ if }x = \frac{1}{n},n \in \mathbb{N}\\ 0, \text{otherwise}\end{cases}$$ and $$g(x) = \begin{cases} n,\text{ if } x = \frac{1}{n}, n \in \mathbb{N} \\ 0, \text{ otherwise}.\end{cases}$$

Then choose the correct option

$a)$ both $f$ and g are Riemann-integrable

$b)$ $f$ is Riemann-integrable but $g$ is not

$c)$ $g$ is Riemann-integrable but $f$ is not

$d)$ neither $f$ nor $g$ is Riemann-integrable

i thinks option $d)$ will correct that is neither $f$ nor $g$ is Riemann-integrable because $f$ and $g$ are not continuous.

Is its True ??

Answer 1

b) is the correct answer. $g$ is not even a bounded function so it is not a Riemann integrable. $f$ is Riemann integrable because it is bounded and continuous almost everywhere.