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Can anyone think of a counter example that for $f_n:[a,b] \to \mathbb{R}$ regulated and $f_n \to f$ pointwise but $f$ is not a regulated function?


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For notational simplicity take $[a,b] = [0,1]$. The function $$ f(x) = \cases{\sin\left(\frac{1}{x}\right) & 0 < x \leq 1 \\ 0 & x = 0}$$ is non-regulated (see definition) because $\lim_{x\rightarrow 0} f(x)$ doesn't exist. Now take $f_n(x) = f(x) \chi_{[1/n, 1]}(x)$, where $\chi$ is the indicator function. We have that $f_n(x) \rightarrow f(x)$ pointwise, since $f_n(0)=f(0)=0$ for all $n$, and $f_n(x)=f(x)$ for sufficiently large $n$ when $x>0$.

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That's a nice example. –  Patrick Apr 4 '12 at 12:39
Very helpful, thank you! –  user26069 Apr 4 '12 at 12:58

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