Yes or No, Real Analysis, continuity, compactness

Question

Am I correct over statements below?

1. The limsup and liminf of the sequence $n^2$ (meaning $1,4,9,16,\dots$) are equal. T

2. Every bounded sequence has at most one convergent subsequence. F

3. Are the following characteristic functions Riemann integrable on the interval $[0,1]$?

   • $\chi_{\left[0,\frac12\right]}$ yes
   • $\chi_{\Bbb Q}$ no
   • $\chi_C$, where $C$ is the Cantor set yes
   • $\chi_{\Bbb R-\Bbb Q}$ no
   • $\chi_{\left\{\frac1n:n\in\Bbb N\right\}}$ no

4. No continuous function $f:\Bbb R\to\Bbb R$ can have a minimum value. (False)

5. Let $I_1\supset I_2\supset I_3\supset\dots$ be a nested sequence of closed intervals in $\Bbb R$ whose lengths form a decreasing sequence converging to $0$. Choose points $a_n\in I_n$ for each $n$. Then the sequence $a_n$ converges, (I think it’s true)

6. Consider a function $f:\Bbb R\to\Bbb R$. Which of the following statements are true?

   • If $f$ is continuous, then it maps every compact set onto a compact set? yes
   • If $f$ maps every compact set onto a compact set, then it is continuous. no
   • If $f$ is continuous, then it maps every connected set onto a connected set? yes
   • Is it true that if $f$ maps every connected set onto a connected set, then it is continuous. no
   • Is it true that if $f$ is continuous, then it maps every open set onto an open set? yes
   • If $f$ maps every open set onto an open set, then it is continuous. yes

(The original image from which this is copied is here.)

Answer 1

The last two parts of (6) are wrong. First, the constant function $f(x)=0$ is continuous, but the only open set that it maps to an open set is $\varnothing$. For the other example, define an equivalence relation $\sim$ on $\Bbb R$ by $x\sim y$ iff $x-y\in\Bbb Q$. For each $x\in\Bbb R$ the $\sim$-equivalence class of $x$ is $x+\Bbb Q=\{x+q:q\in\Bbb Q\}$, which is clearly dense in $\Bbb R$. Since each $\sim$-equivalence class is countable, there are $|\Bbb R|$ of them, so there is a bijection $\varphi$ from $\Bbb R/\sim=\{x+\Bbb Q:x\in\Bbb R\}$, the set of $\sim$-equivalence classes, to $\Bbb R$. Now define

$$f:\Bbb R\to\Bbb R:x\mapsto\varphi(x+\Bbb Q)\;.$$

Every open interval in $\Bbb R$ contains a member of each $\sim$-equivalence class, so $f$ maps each open interval of $\Bbb R$ onto $\Bbb R$, which is an open set. Thus, $f$ takes open sets to open sets, but $f$ is certainly not continuous.

The last part of (3) is also wrong: $\chi_{\left\{\frac1n:n\in\Bbb N\right\}}$ is bounded and has only countably many points of discontinuity, so it’s Riemann integrable.