Let $X$ be the set of functions from $\mathbb{R}$ to $\mathbb{R}$ which can be written as $$ f = \sum_{i=1}^{\infty} f_i I_{[a_i,b_i]}, $$ where $a_i<b_i$, $f_i$ is continuous, but $f$ need not be continuous at $a_i$ (or $b_i$). What can be said about this set of functions (ex do they contain all $L^p$ functions etc...)
More generally, if $X$ is the set of all functions from $\mathbb{R}^k$ to $\mathbb{R}^n$ with at most $\mathfrak{c}$ many discontinuities, does $X$ contain the set of measurable functions?
Let $f$ be a nowhere continuous bounded function on $[0,1]$. This function belongs to $L^p(\Bbb R)$, yet it does not belong to $X$.
Edit: this function needs to be measurable, but the result still holds.