Exercise 11.9

Question

Prove that the function $F$ given by $(96)$ is continuous on $[a,b]$.

Amit Awasthi · Accepted Answer

\begin{proof}
      Let $\varphi$ be a simple function such that $\int_a^b \lvert F - \varphi
      vert < \epsilon/2$. Then the function $\Phi = \int_a^x \phi\,dt$ is
      continuous on $[a,b]$ by Theorem $6.20$ since $\varphi \in \mathscr{R}$.
      Now, if $y$ is such that $\lvert\Phi(y) - \Phi(x)vert < \epsilon/2$,
      then
      \begin{align*}
        &\lvert F(y) - F(x) vert\
        &\qquad= \left\lvert F(y) - \Phi(y) + \Phi(y) - \Phi(x) + F(x)
        ightvert\
        &\qquad< \int_x^y \lvert f(t) - \phi(t) vert\,dt + \epsilon/2 \le
        \epsilon.\qedhere
      \end{align*}
    \end{proof}

Exercise 11.9

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Exercise 11.9

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