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Let $f:\mathbb{R} \to \mathbb{R}$ is a function with these special properties. $f$ is continuous everywhere. $f$ is not smooth (not infinitely differentiable). $f$ is differentiable only finitely many times everywhere. $f$ belongs to $L^p$. For any $k$ belongs to $\mathbb{N}$ let $E$ be the set of all points where $f$ is differentiable exactly $k$ times then $E$ is not dense in any open subset of $\mathbb{R}$.Is such a function feasible? I need suggestions on ways to construct such a special function.


This is the final version. not going to change it further. I apologize for the inconvinience

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    This is not functional analysis. That refers to questions of continuous linear maps. Retagging.2010-11-13
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    @kahen: I didnt intend it to be...it was a mistake due to autocomplete....thanks for the retagging2010-11-13
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    @chandru: thank you for the latex edit2010-11-13
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    To some of us a "special function" is something completely different.2010-11-13
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    What I mean is completely in the opposite sense ! the other extreme !2010-11-13
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    That's called *average*, *usual*, *normal*, *ordinary*, *typical* ... :-)2010-11-13
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    Rajesh - It is NOT ok to change the question!2010-11-13
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    Rajesh - as I have understood it you must add @name: in front of a comment that should be a notification to a user.2010-11-13
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    @Marek: I am not bothered about the terminology now coz i dont fully understand what you are stating. Please explain the with the current form of the question...there was a mix up earlier and i was away for a while.thank you.2010-11-14

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