For each continuity point So this is a question in Analysis, but it appears to relate to an application in Probability Theory.
If $f_1, f_2, f_3, \ldots$ are monotone increasing functions on $[0,1]$ converging in measure to a measurable $f: [0,1] \to \Bbb R$ then show that, for each continuity point $x$ of $f(x)$ $$f_n(x) \longrightarrow f(x) \ \ \ \text{as} \  \ \ n \longrightarrow \infty $$
I don't know how to prove this for every continuity point as I don't even know how it is defined. I know $(f_n)$ converges unif. to $f$ on $[0,1]$, but this fact and the $\epsilon-\delta$ definition will not prove convergence for every continuity point?

Hints: There is a subsequence $f_{n_k}$ converging a.e. to $f$. Given $\delta >0$ there exist points $y,z$ such that $f_{n_k}(y) \to f(y)$, $f_{n_k}(z) \to f(z)$, $y <x<z$ and $z-y<\delta$. Use continuity of $f$ at $x$ to show that $f_{n_k}(x) \to f(x)$. Now apply this to subsequences of $(f_n)$ to show that every subsequence of $(f_n)$ has  a further subsequence which converges to $f(x)$.
