Suppose $f(x+y)=f(x) + f(y)$ for all $x,y \in \mathbb R$ and $f$ is continuous at a point $a \in \mathbb R$. Prove that $f$ is continuous at every $b \in \mathbb R$.
I know that in order to prove continuity we can use the definition that states that $\lim_{x\to b}f(x+y)=f(b)$ then the function is continuous however I do not know how to show that the limit will be $f(b)$ for the function. Thanks in Advance.
It follows that for any $h\in \mathbb{R}$, $f(a+h)-f(a)=f(b)-f(b-h)$, because $f(b)+f(a)=f(a+b)=f(a+h)+f(b-h)$. Then we have that $f(a+h)\to f(a)$ as $h\to 0$ if and only if $f(b-h)\to f(b)$ $h\to 0$.
HINT:
We have $f(x+h)-f(x)=f(h)$ for all $x$ and all $h$ (Just let $y=h$.).
Then, this holds in particular for $x=a$, the point for which we are given that $f$ is continuous.
Use this to show that $f$ is continuous at $0$ by letting $h\to 0$ and noting $f(0)=0$.
Then, show this implies that $f$ is continuous everywhere since $f(b+h)-f(b)=f(a+h)-f(a)=f(h)$.
Take any $b$, then exist $c$ such that $b=c+a$. So for each $\epsilon >0$ exist $\delta >0$ such that if $|h|<\delta $ we have:
$$|f(b+h)-f(b)| = f(a+h)+f(c)-f(a)-f(c)| = |f(a+h)-f(a)|<\epsilon$$
