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I was thinking about the above problem.Can someone point me in the right direction?Thanks everyone in advance for your time.

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up vote 2 down vote accepted

Hint: This function is continuous exactly at the solutions of $\frac{3x}4=\sin x$ (why?)

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@user33640: Note that we have always a sequence of rationals approaching to an irrational number and vice versa. – Babak S. Jan 15 '13 at 18:20
@BabakSorouh Thanks a lot sir.I think i have got it.Then there are infinitely many points where the function is continuous.Am i right? Please confirm. – user52976 Jan 16 '13 at 3:06
@user33640: No, see the picture and tell in how many points the two functions intersect each other. – Babak S. Jan 16 '13 at 4:46
$\frac{3x}4>1\ge\sin x$ as soon as $x>\frac43$ and $\frac{3x}4<-1\le\sin x$ as soon as $x<-\frac43$. Therefore intersections can occur only in $[-\frac43,\frac43]$. – Hagen von Eitzen Jan 16 '13 at 6:25
@user33640: It was a very kind of you that accepted a very small job I did for you, BUT note that all things we could defeat the problem was completely done by Hagen. So, please accept his. In fact, I didn't nothing but a very small piece here. Thanks a lot. :-) – Babak S. Jan 16 '13 at 8:55

enter image description here

They intersect at 3 points each other.

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Thanks a lot sir.I have got it.It is crystal clear now. – user52976 Jan 16 '13 at 5:03
A count of $3$ would be even more crystal clear ... – Hagen von Eitzen Jan 16 '13 at 6:26
Pictures speak more than words, sometimes! +1 – amWhy Feb 16 '13 at 0:15

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