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Integration

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The setting: $ f$ bounded on $ [a,b]$, $ P: a=t_0<t_1<\dots<t_n=b$ a partition of $ [a,b]$, $ m_i=\inf_{[t_{i-1},t_i]}f(x)$, $ M_i=\sup_{[t_{i-1},t_i]}f(x)$, $ L(f,P)=\sum_{i=1}^n m_i(t_i-t_{i-1})$, $ U(f,P)=\sum_{i=1}^n M_i(t_i-t_{i-1})$, $ L(f)=\sup_P L(f,P)$, $ U(f)=\inf_P U(f,P)$. Finally, if $ U(f)=L(f)$ we say that ``$ f$ is integrable on $ [a,b]$'' and set $ \int_a^b f=\int_a^b f(x)dx=U(f)=L(f)$.

Theorem 1. For any two partitions $ P_{1,2}$, $ L(f,P_1)\leq U(f,P_2)$.

Theorem 2. $ f$ is integrable iff for every $ \epsilon>0$ there is a partition $ P$ such that $ U(f,P)-L(f,P)<\epsilon$.

Theorem 3. If $ f$ is continuous on $ [a,b]$ then $ f$ is integrable on $ [a,b]$.

Theorem 4. If $ a<c<b$ then $ \int_a^b f=\int_a^c f+\int_c^b f$ (in particular, the rhs makes sense iff the lhs does).

Theorem 5. If $ f$ and $ g$ are integrable on $ [a,b]$ then so is $ f+g$, and $ \int_a^b f+g = \int_a^b f + \int_a^b g$.

Theorem 6. If $ f$ is integrable on $ [a,b]$ and $ c$ is a constant, then $ cf$ is integrable on $ [a,b]$ and $ \int_a^b cf = c\int_a^b f$.

Theorem $ \mathbf{7^a}$. If $ f\leq g$ on $ [a,b]$ and both are integrable on $ [a,b]$, then $ \int_a^b f\leq\int_a^b g$.

Theorem 7. If $ m\leq f(x)\leq M$ on $ [a,b]$ and $ f$ is integrable on $ [a,b]$ then $ m(b-a)\leq\int_a^b f\leq M(b-a)$.

Theorem 8. If $ f$ is integrable on $ [a,b]$ and $ F$ is defined on $ [a,b]$ by $ F(x)=\int_a^x f$, then $ F$ is continuous on $ [a,b]$.



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The two boxed statements on this page are FALSE.
\end{center}}}          \fbox{\parbox{2.5in}{\begin{center}\large
White unicorns roam the earth.
\end{center}}}

Just for fun. Why did I put these boxed statements on this page? Can they both be true? Can they both be false? If just one is true, which one must it be?

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Dror Bar-Natan 2005-01-05