Which of the following statements is/are true?
(a)Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be such that $f(x+y)=f(x)+f(y)$ for all $x, y \in \mathbb{R}$. If $f(t)>0$ for all $t>0$, then $f$ is continuous.
(b) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be such that $f(x+y)=f(x) f(y)$ for all $x, y \in \mathbb{R}$. Then $f(x) \geq 0$ for all $x \in \mathbb{R}$.
(c) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous at $x=0$ and such that $f(x)+f(x / 2)=0$ for all $x \in \mathbb{R}$. Then $f(x)=0$ for all $x \in \mathbb{R}$.
(d) None of the above
I have been able to prove that (b), and (c) are correct but I am stuck with option (a). The answer says it's true. Any help will be truly appreciated.
