-1
$\begingroup$

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.

  • 0
    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

1 Answers 1

1

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.