If a bounded function is integrable on each interval $[a,1]$, then it is integrable on $[0,1]$.

Question

Let $f:[0,1] \to \mathbb{R}$ bounded,such that $\forall a $ with $0<a<1$, $f$ is integrable at the interval $ [a,1]$,show that f is integrable at $[0,1]$.

As $f$ is bounded, $\exists M>0$ such that $|f(x)| \leq M, \forall x \in [0,1]$.

Let $0<\epsilon<1$ .As $f$ is integrable at $[\epsilon,1]$, $\exists $ partition $P_1$ of $[\epsilon,1]$ ,such that:

$$U(f,P_1)-L(f,P_1)< \epsilon$$

Now,we consider the partition $P_2=\{0\} \cup P_1$ of $[0,1]$.

It is :

$$U(f,P_2)-L(f,P_2)=(\epsilon-0) \cdot \sup f([0, \epsilon])+U(f,P_1)-(\epsilon-0) \cdot \inf f([0, \epsilon])-L(f,P_1)$$

I jut wanted to know if $\sup f([0, \epsilon])=M \text{ and } \inf f([0, \epsilon])=-M$ or if it is just $\sup f([0, \epsilon]) \leq M \text{ and } \inf f([0, \epsilon]) \leq -M$.

Thanks in advance!

Answer 1

Choose $\delta<\text{mini}\{1,\frac{\epsilon}{2},\frac{\epsilon}{4M}\}$. And then choose a partition for the $\delta$. Then we will get $U(P_2,f)-L(P_2,f)\leq 2\delta M<\frac{\epsilon}{2}$ where $P_2=\{0\}$ and $P_1=\{\delta,\ldots,1\}$.