Let $f:[-4,\infty)\rightarrow S$ be a continuous function. Then which of the following is/are true?

Question

Let $f:[-4,\infty)\rightarrow S$ be a continuous function. Then which of the following is/are true?

1. If $S$ is closed, then $f$ has a fixed point.

2. If $S=(-2,\epsilon)$, then $f$ has a fixed point.

3. If For every $\epsilon>0, S=(-4,\epsilon)$, $f$ has a fixed point.

4. If $S=[-1,\infty)$, then $f$ has a fixed point.

I know that $1,4$ are false. To justify this, we can choose $f=x+3$ and $S=[-1,\infty)$. But how to conclude about options $2$ and $3?$

Answer 1

2. Let $g(x)=f(x)-x$. Then $g(-4)=f(-4)+4 >-2+4>0$ and $g(x) \to -\infty$ as $x \to \infty$. Hence, by IVP, there exists $x$ such that $g(x)=0$.

A similar argument works for 3) also.