# Verification of few Riemann integration functions

Pick out the functions which are Riemann integrable on $[0,1]$

(a) $f(x)= 1$ , if $x$ is rational and $f(x)=0$, if $x$ is irrational
(b) $f(x)= 1$ , if $x∈{\alpha_1, \alpha_2, ….,\alpha_n}$ is rational and $f(x)=0$, otherwise, where $\alpha_1, \alpha_2, ….,\alpha_n$ are fixed , but arbitrarily chosen numbers in $[0,1]$
(c) and $f(x)=0$, if $x$ is irrational or if $x=0$ And $f(x)=\sin(q\pi)$, if $x=\frac{p}{q}$ , and $q$ positive and coprime integers.
(d) $f(x) = \lim_{n\to\infty} \cos^{2n}(24\pi x)$.
(e) $f(x)= \cos x$ , if $0≤x≤\frac{1}{2}$ and $f(x)=\sin x$, if $\frac{1}{2}<x≤1$

(a) is not true and it it a very common example.
(b) since only finitely many dis continuities so it is true.
(c) no idea
(d) no idea
(e) true. Since only one point of discontinuity.

For ($c$), simply note that $\sin(q\pi)=0$ for all integers $q$, hence $f(x)=0$ for all $x\in[0,1]$. For $(d)$, recall that $\lim_{n\to\infty}x^n=0$ for all $x\in[0,1)$. What is the range of $\cos^{2n}(24\pi x)$ and when does it equal the maximum? This should provide you with enough information to answer the question.