Suppose f:[a,b] $\rightarrow$ $\mathbb{R}$ is a bounded function and there is a set Z $\subset$ [a,b] sucht that:
- f is continuous at every point x $\notin$Z.
- For every $\epsilon$ > 0, the set Z can be covered by finitely many intervals with total length less than $\epsilon$.
Show that f is Riemann integrable on [a,b]
Have no idea how to handle this proof? Any hints or suggestions.
Hint: Once you cover Z by a set U which is a union of open intervals of total length $<\epsilon$, f is uniformly continuous on $[a, b] - U$.