- #1

Eclair_de_XII

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- TL;DR Summary
- -continuities.

Define ##f:[a,b]\longrightarrow \mathbb{R}##. Assume that ##f## is non-negative and bounded. Suppose there exists a point ##y_0\in [a,b]## at which ##f## fails to be continuous and suppose also that there exists a sequence of points ##a_n\in[a,b]## that converge to ##y_0##, where ##f## fails to be continuous at each ##a_n##. Then ##f## is integrable.

Assume any function ##f## with only finitely many discontinuities is integrable.

We show that there is a partition s.t. the upper sum and the lower sum of ##f## w.r.t. this partition converge onto one another.

Let ##\epsilon>0##.

Define a sequence of functions ##g_n:[a,b]\setminus(\{a_n\}_{n\in\mathbb{N}}\cup\{y_0\})## s.t. ##g_n(x)=|f(x)-f(a_n)|##. Suppose there is a ##g_m## that is not bounded. Then there must exist a point ##x'## s.t. ##|f(x')-f(a_m)|>\sup f[a,b]##. This is a contradiction. Hence, each ##g_n## is bounded, and moreover, ##G:=\{g_n(x):x\in\textrm{dom}(g_n)\}_{n\in\mathbb{N}}## is bounded.

Set ##\alpha=\sup G## and set ##\delta=\frac{1}{4}\cdot\frac{\epsilon}{\alpha}##. Now define points ##t_1,t_2## to be the endpoints of ##B(y_0,\delta)##. Set ##M=\sup f(B(y_0,\delta))## and ##m=\inf f(B(y_0,\delta))##; set ##\Delta t=t_2-t_1## also. Then the following must hold:

\begin{eqnarray}

M\Delta t - m\Delta t &=&(M-m)\Delta t \\

&\leq&\alpha\Delta t \\

&=&2\delta\alpha\\

&=&2\left(\frac{1}{4}\cdot\frac{\epsilon}{\alpha}\right)\cdot\alpha \\

&=&\frac{\epsilon}{2}

\end{eqnarray}

Choose a partition ##P:=\{x_0,\ldots,x_n\}## for ##[a,b]\setminus B(y_0,\delta)## s.t. ##U(P,f)-L(P,f)<\frac{\epsilon}{2}##. Now refine the partition with the points ##t_i## as described above in order to yield the result.

Let ##\epsilon>0##.

Define a sequence of functions ##g_n:[a,b]\setminus(\{a_n\}_{n\in\mathbb{N}}\cup\{y_0\})## s.t. ##g_n(x)=|f(x)-f(a_n)|##. Suppose there is a ##g_m## that is not bounded. Then there must exist a point ##x'## s.t. ##|f(x')-f(a_m)|>\sup f[a,b]##. This is a contradiction. Hence, each ##g_n## is bounded, and moreover, ##G:=\{g_n(x):x\in\textrm{dom}(g_n)\}_{n\in\mathbb{N}}## is bounded.

Set ##\alpha=\sup G## and set ##\delta=\frac{1}{4}\cdot\frac{\epsilon}{\alpha}##. Now define points ##t_1,t_2## to be the endpoints of ##B(y_0,\delta)##. Set ##M=\sup f(B(y_0,\delta))## and ##m=\inf f(B(y_0,\delta))##; set ##\Delta t=t_2-t_1## also. Then the following must hold:

\begin{eqnarray}

M\Delta t - m\Delta t &=&(M-m)\Delta t \\

&\leq&\alpha\Delta t \\

&=&2\delta\alpha\\

&=&2\left(\frac{1}{4}\cdot\frac{\epsilon}{\alpha}\right)\cdot\alpha \\

&=&\frac{\epsilon}{2}

\end{eqnarray}

Choose a partition ##P:=\{x_0,\ldots,x_n\}## for ##[a,b]\setminus B(y_0,\delta)## s.t. ##U(P,f)-L(P,f)<\frac{\epsilon}{2}##. Now refine the partition with the points ##t_i## as described above in order to yield the result.