Determine function which is continuous at a point

Question

Let $f:\mathbb{R} \rightarrow \mathbb{R}$, be continuous at $\pi$ and satisfy $f(x + y) = f(x) + f(y)$ for all $x,y \in \mathbb{R}$. Determine $f.$

Answer 1

If I have understood correctly the question you are asking to find a function which satisfy that property.

I would say that $f:\mathbb{R}\to\mathbb{R}$ defined as $f(x)=x$ for each $x\in\mathbb{R}$ works since $f(\pi)=\pi$ and $f(x+y)=x+y=f(x)+f(y)$ for each $x,y\in\mathbb{R}$.