$f:\Bbb R\to \Bbb R$ be a continuous function such that $f(i)=0\forall i\in \Bbb Z$

Question

Let $f:\Bbb R\to \Bbb R$ be a continuous function such that $f(i)=0\forall i\in \Bbb Z$.

Which of the following is always true:

1. $Image(f)$ is closed in $\Bbb R$.
2. $Image(f)$ is open in $\Bbb R$.
3. $f$ is uniformly continuous.

My try:

1.I am unable to find a example here.

2. False take $f(x)=0$.

3.I am unable to find a example here.

Please give some hints here.

Answer 1

As regards 1) consider the function $$f(x)=\arctan(x)\cdot\sin(\pi x)$$ What is $f(\mathbb{R})$?

For iii) take $f(x)=\sin(\pi x^2)$. Is it uniformly continuous in $\mathbb{R}$?