Exercise 9.7

Question

Exercise 7: 
    Suppose that $f$ is a real-valued function defined in an open set $E\subset\R n$, and that the partial derivatives $D_1f,\ldots,D_nf$ are bounded in $E$. Prove that $f$ is continuous in $E$.

analambanomenos · Accepted Answer

\def\eps{\varepsilon}
\def\ints{\mathbb{Z}}

Suppose that $|D_jf|<M$ in $E$, for $j=1,\ldots,n$. Following the hint to mimic the proof of Theorem 8.21, fix $\v x\in E$ and let $\varepsilon>0$. Since $E$ is open, there is an open ball 
    $S\subset E$, with center at $\v x$ and radius $r<(Mn)^{-1}$. Suppose $\v h=\sum h_j\v e_j$, $|\v h|<r$, put $\v v_0=\v 0$, and $\v v_k=h_1\v e_1+\cdots+h_k\v e_k$, for $1\le k\le n$. Then
    $$
        f(\v x+\v h)-f(\v x)=\sum_{j=1}^n\big(f(\v x+\v v_j)-f(\v x+\v v_{j-1})\big). \leqno{(*)}
    $$
    Since $| \v v_k|<r$ for $1\le k\le n$ and since $S$ is convex, the segments with endpoints $\v x+\v v_{j-1}$ and $\v x+\v v_j$ lie in $S$. Since $\v v_j=\v v_{j-1}+h_j\v e_j$, the mean value
    theorem, Theorem 5.10, shows that the $j$th summand in $(*)$ is equal to
    $$
        h_j(D_jf)(\v x+\v v_{j-1}+\theta_jh_j\v e_j)
    $$
    for some $\theta_j\in(0,1)$. By $(*)$, it follows that 
    \begin{align*}
        \big|f(\v x+\v h)-f(\v x)\big| &\le \sum_{j=1}^n\big|f(\v x+\v v_j)-f(\v x+\v v_{j-1})\big| \
        &= \sum_{j=1}^n\big|h_j(D_jf)(\v x+\v v_{j-1}+\theta_jh_j\v e_j)\big| \ 
        &< \sum_{j=1}^n|h_j|M<(Mn)r<\varepsilon
    \end{align*}
    which shows that $f$ is continuous at $\v x$. Since $\v x\in E$ was arbitrary, we have $f$ continuous on $E$.

Exercise 9.7

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Exercise 9.7

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