# Odd bounded function limit at $x=0$

$$f:\mathbb{R} \to \mathbb{R}$$ is a bounded odd function. What can be said about $$\lim_{x \to 0}f(x)$$?

1. Must be $$0$$
2. If exists then $$0$$
3. Must exist but may not be $$0$$

I am stuck at this question. I know, for continuous odd function we must have value $$0$$ at $$x=0$$. But we don't know if this function is continuous or not. How to proceed? Thanks in advance.

$$\lim_{x\to0^-}f(x)=-\lim_{x\to0^+}f(x).$$
• @user587126 Take the function which is $0$ at $x = 0$, $1$ on $\mathbb R^+$, and $-1$ on $\mathbb R^-$. – Alex Provost Jul 18 '19 at 12:29
• @user587126 Consider $$f(x) = \begin{cases} 1, &\text{ if x\notin \mathbb{Q}, x > 0}\\ -1, &\text{ if x\notin \mathbb{Q}, x < 0}\\ 0, &\text{ if x\in \mathbb{Q}}\\ \end{cases}$$ Then $\lim_{x\to 0^+} f(x)$ and $\lim_{x\to 0^-} f(x)$ both don't exist. – mechanodroid Jul 18 '19 at 12:59