few real analysis true/false questions. Which of the following statements are true and why?


*

*Any continuous function from the open unit interval $(0, 1)$ to itself has a fixed
point.

*$\log x$ is uniformly continuous on $( 1/2,+\infty)$.

*If $A, B$ are closed subsets of $[0,\infty)\,$, then $A + B = \{x + y\; |\; x \in A,\, y \in B\}$
is closed in $[0,\infty)$

*A bounded continuous function on $\mathbb{R}$ is uniformly continuous.

*Suppose $f_n(x)$ is a sequence of continuous functions on the closed interval $[0, 1]$
converging to $0$ pointwise. Then the integral $\int_0^1f_n(x)\mathrm dx\,$ converges to $0$.
My thoughts:


*

*I am not sure asthe interval is not closed.

*It is true as it has bounded derivative.

*Usually, the sum of two closed set is not closed but is the case here?

*Not sure.

*Not sure.


Can anyone help me please to solve the problems? Thank you.
 A: *

*Define $f(x)=\sqrt{x}$, for every $x \in (0,1)$. Then ...

*Join the point $(0,n)$ and the point $(1/n,0)$ with a segment, then join $(1/n,0)$ and $(1,0)$ with a horizontal segment. This is the graph of a function $f_n$ such that $f_n \to 0$ pointwise and $\int_0^1 f_n(x)\, dx = 1/2$ for each $n$.
A: *

*$x^{2}$ is good too.

*You are right.

*Let $z_{n}\in A+B$ such that $z_{n}\rightarrow z$. So $z_{n}=a_{n}+b_{n}$ with $a_{n}\in A$ and $b_{n}\in B$. If $b_{n}$ or $a_{n}$ is unlimited we get the absurd (note that $z_{n}$ converges) of $a_{n}+b_{n}$ being unlimited, because  $b_{n}\geq 0$ and $a_{n}\geq 0$. So $a_{n}$ and $b_{n}$ are limiteds and then you can extract subsequences that are convergents. Now you can conclude.

*Take the function $$x \mapsto \sin(x^2)$$ This function is continuous, limited, but not uniformly continuous. Can u see it?

A: *

*Or $f(x) = x^n$ or $x^{\frac{1}{n}}$.

*Yes. Lipschitz continuity implies uniform continuity and $|f'| = |\frac1x| \leq 2$.

*Yes. It is true for intervals: $+([a,b] \times [c,d]) = [a+c, b+d]$. Hence it is true for all closed sets.

*To add to Kaye's answer: to "break" uniform continuity (while still being continuous) you are looking for a bounded continuous function with unbounded derivative. That is, you need your function to become arbitrarily steep in places. $x \mapsto \sin x^2$ has this property (check it!).
5.Define $$ f_n(x) = \begin{cases}
n^2 x - n & x \in [\frac{1}{n}, \frac{2}{n}] \\ 
- n^2 x + 3n & x \in [\frac{2}{n}, \frac{3}{n}] \\
\end{cases}$$ for $n \geq 3$. This looks like this: 

(drawn using slimber)
