# Which of the followings are true (J.N.U)?

Consider $f : (0 , 1) \rightarrow [0,1]$ be a continous function . Which of the followings are true ?

1. $f$ is uniformly continous.

2. $f$ has a fixed point.

3. $f$ is a differentiable function.

4. There is a continous function $\overline f$ on $[0,1]$ such that $\overline f = f$ on $(0,1)$

I have tried :

1) suppose $f(x) =\frac{1}{x}$ is continous but not uniformly continous

3) Suppose $f(x) = |x-\frac{1}{2} |$ which is continous but not differentiable.

4) If $f$ is a contnous function on $E \subset \mathbb R$, then there exist a continous function $g$ on $\mathbb R$ such that $f = g$ on $E$ if $E$ is closed on $\mathbb R$. I think we can use this result, but i have no counter example.

$f(x) = 1/x$ is not a valid counter-example for (1) because it does not map $(0,1)$ into $[0, 1]$. But you can choose $f(x) = \sin(1/x)$ instead. That also serves as counterexample for (4) because $\lim_{x \to 0} f(x)$ does not exist.
$f(x) = x/2$ is a counterexample for (2). (This works only because $f$ is defined on an open interval. A continuous function $F: [0,1] \to [0,1]$ necessarily has a fixed-point, this follows from the intermediate value theorem.)