# Proof of Generalized Riemann Integrability Criterion [closed]

Suppose f:[a,b] $$\rightarrow$$ $$\mathbb{R}$$ is a bounded function and there is a set Z $$\subset$$ [a,b] sucht that:

1. f is continuous at every point x $$\notin$$Z.
2. 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.

## closed as off-topic by RRL, Alexander Gruber♦Apr 28 at 8:19

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• @KaviRamaMurthy -- It seems to be only a special case of Lebesgue's Criterion, as the the covers of the set $Z$ are finite in the OP's formulation, whereas a set of Lebesgue measure zero allows countable, infinite covers. – uniquesolution Apr 24 at 6:59

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 $$[a, b] - U$$.