Let $A\subseteq\mathbb R^n$ be a box, let $f: A\to \mathbb R$ be a bounded function.
Suppose there exists a set $B \subseteq A$ such that $B$ is Riemann measurable and has volume zero, such that $f$ is continuous on $A\setminus B$.
Then $f$ is Riemann integrable.
I understand this, but I can't find a formal proof of it.
I know how I would prove that if $A\subseteq \mathbb R^n$ was a Riemann measurable set then every continuous bounded function $f:A \to R$ is Riemann integratable.
Is there a way to adapt this proof?