Exercise 6.4.10

Question

Let $\left\{r_{1}, r_{2}, r_{3}, \ldotsight\}$ be an enumeration of the set of rational numbers. For each $r_{n} \in \mathbf{Q}$, define
  $$
  u_{n}(x)= \begin{cases}1 / 2^{n} & 	ext { for } x>r_{n} \ 0 & 	ext { for } x \leq r_{n} .\end{cases}
  $$
  Now, let $h(x)=\sum_{n=1}^{\infty} u_{n}(x)$. Prove that $h$ is a monotone function defined on all of $\mathbf{R}$ that is continuous at every irrational point.

Ulisse Mini · Accepted Answer

Using $M_n = 1/2^n$, by the Weierstrass M-Test we have that $h$ converges uniformly. Since each $u_n$ is continuous at all irrational numbers, we have that $h$ is continuous at all irrational numbers. Monotone-ness comes from applying the Order Limit Theorem to compare the series $\sum u_n(a)$ and $\sum u_n(b)$ for $a,b\in \mathbf{R}$.

Exercise 6.4.10

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Exercise 6.4.10

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