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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.

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

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  • $\begingroup$ @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. $\endgroup$ – uniquesolution Apr 24 at 6:59
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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 $[a, b] - U$.

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