Example of sequence of continuous function which is not Riemann integrable with following property

$$\{f_n\}$$ is a sequence of continuous functions with following properties:

1. $$0 \leq f_n(x) \leq 1, \forall x \in \Bbb R$$
2. $$f_n(x)$$ is monotonically decreasing sequnce as $$n\to \infty$$
3. The Limiting function $$f$$ is not Riemann integrable.

I know that such function is Lebesgue integrable using Monotone convergence theorem but not sure about Riemann integrable.

If there is counterexample please give me that. Or give me a hint so that I can prove this theorem

Any help will be appreciated.

• I think you can take the counter-example from here math.stackexchange.com/questions/612098/… – Yanko Mar 10 at 13:52
• Dear Sir, That $f_n$ is not continuous also not monotonically decreasing – SRJ Mar 10 at 14:00
• Oh yeah I didn't see you want $f_n$ to be continuous. (You could work out the monotonically decreasing by taking $1-f_n$). – Yanko Mar 10 at 14:01
• Now I think I'm not wrong that the answer is in the comment here math.stackexchange.com/questions/2670955/… (open the wikipedia page, 4th bullet) – Yanko Mar 10 at 14:09
• By 2 do you mean $f_1(x) \ge f_2(x) \ge \cdots$ for each $x?$ – zhw. Mar 10 at 17:47