Exercise 7.8

Question

\def\ints{\mathbb{Z}}
Exercise 8: 
    If $$I(x)=
    \begin{cases}
        0 & (x\le 0), \
        1 & (x>0),
    \end{cases}$$
    if $\{x_n\}$ is sequence of distinct points of $(a,b)$, and if $\sum|c_n|$ converges, prove that the series $$f(x)=\sum_{n=1}^\infty c_nI(x-x_n)\quad\quad(a\le x\le b)$$ converges uniformly,
    and that $f$ is continuous for every $x
eq x_n$.

analambanomenos · Accepted Answer

\def\eps{\varepsilon}

Let $f_n(x)$ be the partial sum $\sum_{k=1}^nc_kI(x-x_k)$. Then $$\big|f_n(x)-f_m(x)\big|\le\sum_{k=m+1}^n\big|c_kI(x-x_k)\big|\le\sum_{k=m+1}^n|c_k|.$$ Since $\sum|c_k|$ converges, by
    Theorem 3.22 for any $\varepsilon>0$ there is an integer $N$ such that for $m\ge N$ and $n\ge N$ we have $$\big|f_n(x)-f_m(x)\big|\le\sum_{k=m+1}^n|c_k|<\varepsilon.$$ Hence by Theorem 7.8
    the series converges uniformly.

Let $x\in(a,b)$ such that $x
e x_n$ for any $n$. Then the partial sum $f_n$ is constant in any neighborhood of $x$ not containing any of $x_1,\ldots,x_n$. Hence by Theorem 7.11,
    $$\lim_{tightarrow x}f(x)=\lim_{nightarrow\infty}f_n(x)=f(x),$$ that is, $f$ is continuous at $x$.

Exercise 7.8

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Exercise 7.8

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