Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?


share|cite|improve this question
up vote 3 down vote accepted

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 $x>0$.

share|cite|improve this answer
That's a nice example. – Patrick Apr 4 '12 at 12:39
Very helpful, thank you! – user26069 Apr 4 '12 at 12:58

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.