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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.

Can anybody help me please.

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up vote 0 down vote accepted

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.

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then both the functions are true in c and d . am i right –  poton Jan 1 '13 at 3:20
    
That is correct @poton. –  Clayton Jan 1 '13 at 3:22
    
are my views correct for the other options? –  poton Jan 1 '13 at 3:26
    
Yes, your other answers are correct. –  Clayton Jan 1 '13 at 3:29
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