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One can construct Riemann Integrable functions on [0,1], that have a dense set of discontinuities as follows.

(a) Let $f(x) =0$ when $x < 0$, and $f(x) = 1$ if $x\geq$ 0. Choose a countable dense sequence ${r_n}$ in [0,1]. Then Show that the function $$F(x) = \sum_{n = 1}^{\infty}\frac{1}{n^2}f(x - r_n)$$. is integrable and has discontinuities at all the points of the sequence ${r_n}$.Do I have to show that $F$ is monotonic and bounded? and the question has parts(b) & (c) also, Do part (a) is a way to construct Riemann Integrable functions on [0,1], that have a dense set of discontinuities or part (a) combined with part (b) & (c)?

(b)Consider next$$F(x) = \sum_{n=1}^{\infty}3^{-n} g(x- r_{n})$$. where $g(x)= sin\frac{1}{x}$ when $x\ne 0$, and $g(0) = 0$. Then $f$ is integrable, discontinous at each $x = r_n$, and fails to be monotonic in any subinterval of [0,1].

Could anyone help me?

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    Part $(a)$ alone asserts that the specific function $F:I \to \mathbb{R}$ is Riemann Integrable on the unit interval yet discontinuous on the dense subset $E = \{r_n:n\} \subset I$, but you have to prove the validity of this assertion (Note $F$ is integrable because it is the uniform limit of Riemann integrable functions. I would fix some $r_{n_0}$ and see how you can show $F$ is discontinuous at this fixed point). Also, e don't know what parts $(b),(c)$ are. – 2017-02-28
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    Okay, I will complete the question and write parts(b) and (c). Do I have to prove the validity of these assertion usind what u stated between brackets?. thanks. – 2017-02-28
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    You don't *have to* - I just made a suggestion. – 2017-02-28
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    .Do I have to show that $F$ is monotonic and bounded? – 2017-03-01
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    This may help http://math.stackexchange.com/questions/1700207/how-to-prove-this-function-is-integrable?rq=1 – 2017-03-01
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    I had found it from about 45 minuites, :) anyway thank u very much. Is there a relation between part(a)&(b)? – 2017-03-01

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