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First of all sorry if this has been posted before, I had look but couldn't find anything thats exactly like my question. Suggest I am given a function $f[a,b] \rightarrow \mathbb{R} $that is regulated, Then I know that at any given point its has a left hand and right hand limit (unless its a or b in which case just one of them do). Can I go on to say that any sequence $a_n$ that is bounded above by a number $x$ has the property that $lim_{n \rightarrow \infty} f(a_n) = f(x_-)$ and a similiar thing for bounded below? Can I even use sequential continuity on bounded domain?

Thanks

Henry

  • Indeed, more is true: for any function $f$, $\lim_{x \to a^-} f(x)$ exists and equals $L$ if and only if for every sequence $x_n < a$ that converges to $a$, we have $\lim_{n \to \infty} f(x_n) = L$. You might try proving this from the definitions. – Nate Eldredge Oct 20 '13 at 19:24
  • What do you mean by "regulated"? – Cameron Buie Oct 20 '13 at 19:25
  • Thanks for that Nate! And by regulated I mean for any $\epsilon > 0$ there exists some step function $\phi$ such that $||\phi - f||_\infty$ (uniform norm) $< \epsilon$. Hope that helps –  Oct 20 '13 at 19:33

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