One of the most strange things that I learned in Real Analysis is a function that is continuous on $\mathbb{R}$ but that is not differentiable everywhere.
I wonder how to prove that no monotone function on $(a,b)$ can be not differentiable everywhere.
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.
any monotone function function is almost continuous everywhere and so there is no need to assume continuity
I couldn't prove or disprove that claim but my intuition was: If a monotone function is not differentiable at $x$ then there must be some Jump i.e $\lim\limits_{x\to c^-}f(x)-\lim\limits_{x\to c^+}f(x)\ne0$ so there must be uncountable many Jumps and uncountable series of positive numbers must diverge (I couldn't find a rigorous proof for these two claims) so that mean $f(x)$ don't exist.
