Exercise 6.4

Question

Exercise 4: 
    If $f(x)=0$ for all irrational $x$, $f(x)=1$ for all rational $x$, prove that $f
otin\mathscr R$ on $[a,b]$ for any $a<b$.

ghostofgarborg · Accepted Answer

(	extit{Matt ``Frito'' Lundy}) \
    For any partition $P$, we have
    $$U(P,f) = b-a > 0$$
    $$L(P,f) = 0,$$
    so $f 
otin \mathscr R$.

Amit Awasthi · Answer

\begin{proof}
      Let our partition $P$ be as follows
      \begin{equation*}
        P = \{a=x_1,x_2,\ldots,x_{n-1},x_n = b\}
      \end{equation*}
      where $x_{n-1} < x_n$ for all $n$. Then,
      \begin{align*}
        U(P,f,\alpha) &= \sum_{i=1}^n M_i \Delta \alpha_i = b-a,\
        L(P,f,\alpha) &= \sum_{i=1}^n m_i \Delta \alpha_i = 0,
      \end{align*}
      for $P$, and all refinements of $P$, since both the rationals and the irrationals are dense in $\mathbb{R}$, and therefore any subinterval of $[a,b]$ would contain both a rational and an irrational.
\end{proof}

Exercise 6.4

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Exercise 6.4

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