$X=C^1([0, 1])$ and $Y=C([0, 1])$, both with sup norm. Define $F: X \rightarrow Y$ by $F(x)=x+x'$, ($x'$ is sign of derivative) and $f(x)=x(1)+x'(1)$ for $x \in X$
Then which of the following is true?
1) $F$ is continuous map.
2) $F$ is closed map.
3) $f$ is continuous map.
4) $f$ is closed map.
My work: If we take $x_n(t)=t^n$ then $\vert \vert F \vert \vert=1+n$ which is unbounded as $n \rightarrow \infty$.So $F$ is not continuous.
But $\vert f(x) \vert \leq 2 \vert \vert x \vert \vert_{\infty}$, so $f$ is continuous and as continuous map is closed map $f$ is also a closed map.
What about closedness of $F?$, Check the above facts and please help.