Exercise 11.12

Question

Suppose
    \begin{enumerate}[label=(\alph*)]
      \item $\lvert f(x,y) \rvert \le 1$ if $0 \le x \le 1$, $0 \le y \le 1$,
      \item for fixed $x$, $f(x,y)$ is a continuous function of $y$,
      \item for fixed $y$, $f(x,y)$ is a continuous function of $x$.
    \end{enumerate}
    Put
    \begin{equation*}
      g(x) = \int_0^1 f(x,y)\,dy \qquad (0 \le x \le 1).
    \end{equation*}
    Is $g$ continuous?

Amit Awasthi · Accepted Answer

\begin{proof}
      We have, for any sequence $x_n 	o x$,
      \begin{equation*}
        g(x_n) = \int_0^1 f(x_n,y)\,dy.
      \end{equation*}
      By $(a)$, the dominated convergence theorem applies, giving
      \begin{multline*}
        \lim_{n 	o \infty} g(x_n) = \int_0^1 \lim_{n	o\infty}f(x_n,y)\,dy\
        = \int_0^1 f(x,y)\,dy = g(x),
      \end{multline*}
      i.e., $g$ is continuous.
    \end{proof}

Exercise 11.12

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Exercise 11.12

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