Show, using the following theorem:
(Theorem) Let $I\subseteq\mathbb{R}$ be an interval and let $f:I\to\mathbb{R}$ be monotone on $I$. Then the set of points $D\subseteq I$ at which $f$ is discontinuous is a countable set."
that if $f$ is a monotone function satisfying the functional equation $f(x+y)=f(x)+f(y)$ for all $x,y\in\mathbb{R}$, then $f$ must be continuous on $\mathbb{R}$.
I have already proved the another problem that if $f$ is continuous at $x=0$, and satisfies the functional equation $f(x+y)=f(x)+f(y)$ for all $x,y\in\mathbb{R}$, then $f$ must be continuous on $\mathbb{R}$.
I solved the above problem in a similar way in case of $f$ is continuous at $x=0$, but I could not solve in case of $f$ is monotone on $\mathbb{R}$ using the above theorem.
Give me some hint or advice. Thank you!
Since any interval $I=[a, b]$ is uncountable, you can find a point $c\in I$ such that $f$ is continuous at $c$. Now from $\lim_{x\to c}f(x) = f(c)$, we have $\lim_{x\to 0} f(x) = \lim_{x\to c}f(x-c) = \lim_{x\to c}f(x)-f(c)=f(c)-f(c)=0=f(0)$, hence it is continuous at $x=0$, too.