Suppose $ B \subseteq \mathbb{R} $ and $ f: B \to \mathbb{R} $ is an increasing function. Prove that $ f $ is continuous at every element of $ B $ except for a countable subset of $ B $.
Please provide a proof showing that the set of discontinuities of ( f ) is countable. I couldn't solve it.
I tried to show it by contradiction. I.e. I assumed the number of jumps to be uncountably many and tried to show some contradiction such as at every point the value of the function should go to infinity. but I couldnt prove it properly.