这是一份umass麻省大学 MATH 471作业代写的成功案例
问题 1.
Hence
$$
L(r f)=r L(f)=r U(f)=U(r f) .
$$
On the other hand, if $r<0$, then for any partition $P$ of $[a, b]$, we see that
$$
L(P, r f)=r U(P, f) \quad \text { and } \quad U(P, r f)=r L(P, f)
$$
证明 .
and so
$$
L(r f)=r U(f)=r L(f)=U(r f) .
$$
In both the cases, we see that $r f$ is integrable and
$$
\int_{a}^{b}(r f)(x) d x=r \int_{a}^{b} f(x) d x .
$$
MMATH 471 COURSE NOTES ：
Hence
$$
F(x)-F(a)=\sum_{i=1}^{n}\left[F\left(x_{i}\right)-F\left(x_{i-1}\right)\right]=\sum_{i=1}^{n} g\left(s_{i}\right)\left(x_{i}-x_{i-1}\right)
$$
and so
$$
L\left(P_{e}, g\right) \leq F(x)-F(a) \leq U\left(P_{e}, g\right)
$$
Since we also have
$$
L\left(P_{\epsilon}, g\right) \leq \int_{a}^{x} g(t) d t \leq U\left(P_{\epsilon}, g\right)
$$