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Let $f_n : (a,b) \to \mathbb{R}$ be functions that have finite number of maxima and minima, for $n = 1,2,3...$. Let D be a countable dense subset of $(a,b)$. If sequence $\{f_n\}$ converges to $f$ such that the convergence is non-uniform at the points $x \in D$ and uniform at the points $x \in (a,b)\setminus D$, then does it imply that $f(x) = 0 \forall x \in (a,b)\setminus D$ ?

EDIT : The question didn't come out as what i expected. So I am posting a new one. sorry for the inconvinience.

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    "uniform at the points $x \in (a,b)\setminus D$" seems to be a bit of an unfortunate formulation. Do you mean to say "uniformly on $(a,b) \setminus D$"?2011-05-13
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    @Theo : there seems to be a problem2011-05-13
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    @Rajesh: Does *non-uniform* mean *pointwise* convergence2011-05-13
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    Seems I posted my comment as an answer. Anyway, @Theo, do you understand "non-uniform" here? Pointwise but not uniform?2011-05-13
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    @Theo, @Glen : I will reformulate with your permission...in few minutes2011-05-13
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    @Rajesh and violate my beautiful answer! How rude :D.2011-05-13
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    @Glen : In view of Theo's comments the question doesn'tseems meaningful.2011-05-13
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    @chandru1 : yes thats what i intended .2011-05-13
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    @Glen : request you to permit me or suggest me if i could post it as another question.2011-05-13
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    I basically want discontinuities for $f$ at all $x \in D$.2011-05-13
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    @Rajesh I think it's fine to post another question. Also, if you want the limit to be discontinuous on $D$, then it is perhaps most direct to simply state that as an condition in your new question.2011-05-13
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    Please permit me to delete this question. I know its faulty and i don't want to take downvotes. Please let me know how i can delete this question2011-05-14

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Certainly not. Perhaps you might want to further quantify what you mean by "non-uniform". Note that the sequence of functions $f_i(x) = c$ where $c$ is a fixed non-zero number converges pointwise (as well as uniformly, of course), and is a counterexample.