If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f (x + y) = f (x) f (y)$ and continuous at $0$, then continuous everywhere

One easier thing to do is to notice that $f(x)=(f(x/2))^2$ so $f$ is positive, and assume that it is never zero, since then the function is identically zero. Then you can define $g(x)=\ln f(x)$ and this function $g$ will satisfy the Cauchy functional equation $$ g(x+y)=g(x)+g(y)$$ and the theory for this functional equation is well known, and it is easy to see that $g$ is continuous if and only if it is continuous at $0$.

If this is for a first year real analysis class, I think you might get dinged for not at least saying about the zero case that it's a constant function, and constant functions are continuous. Also, you probably should learn your favorite form of LaTeX; graders tend to like this sort of thing, and happy graders tends to mean better scores, in my experience.

Other than that, excellent work.

Use sequential criterion for continuity of a function at a point and the additive property of a function to show f(x) is continuous for an arbitrary x€R