I saw this problem : Suppose $f$ is a real function defined on $\mathbb{R}^1$ which satisfies $\lim\limits_{h\to0 } f(x+h)- f(x-h)= 0$ for every $x \in \mathbb{R}^1$ Does this imply that $f$ is continuous?
I tried for an hour to prove that $f$ is continuous and failed to (I was very upset because I thought that this function is continuous and I was not able to prove that.) I decided to see the solution and to my surprise the function is not continuous.
That made me wonder how much discontinuities could this function have and I made a claim that is at most countable. I tried to prove this claim but failed to. I hope that this claim is not false (not the same way my first intuition was false )
