1. **State the problem:** We have a function $f:\mathbb{R} \to \mathbb{R}$ that is continuous at $x=\pi$ and satisfies the functional equation $$f(x+y) = f(x) + f(y)$$ for all real numbers $x$ and $y$. 2. **Analyze the functional equation:** The equation $$f(x+y) = f(x) + f(y)$$ is known as Cauchy's functional equation. 3. **Use continuity:** It is a classical result that if a function satisfies Cauchy's equation and is continuous at even a single point (here at $x=\pi$), then the function must be linear. That is, there exists a constant $c$ such that $$f(x) = cx$$ for all $x \in \mathbb{R}$. 4. **Conclusion:** The function $f$ is a linear function passing through the origin, and the constant $c = \frac{f(\pi)}{\pi}$. **Final answer:** $$f(x) = cx \text{ for some constant } c \in \mathbb{R}.$$