Is there an integrable function mapping a closed interval to an open interval

Question

Is there a function $f$ that is integrable on a closed interval $[a,b]$ for $a, b \in \mathbb{R}$ and maps this interval to an open but bounded interval?

Edit:

I should have specified better. What I wanted was more of a function $f$ with a primitive function $F$ (that is $F' = f$ on $(a, b)$ and the one-sided limits of $F'$ are $f(a)$ resp $f(b)$).

score 4 · Answer 1 · answered May 11 '20 at 09:43

4

Sure. $f(0)=f(2)=1$, and $f(x)=x$ for $0 \lt x \lt 2$.

answered May 11 '20 at 09:43

Robert Shore

23,332

1

Thank you. This indeed answers my question (as stated) but not as meant. – kyticka May 11 '20 at 10:33

score 2 · Answer 2 · answered May 11 '20 at 09:43

2

What about $f:[0,2]\to(0,1)$ defined as $f(0)=f(1)=f(2)=1/2$, $f(x)=x$ in $(0,1)$ and $f(x)=2-x$ on $(1,2)$?

answered May 11 '20 at 09:43

GReyes

16,446
11
16

Thank you. This indeed answers my question. But I should have specified better. – kyticka May 11 '20 at 10:33

Is there an integrable function mapping a closed interval to an open interval

2 Answers2