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As the question states, if a function is continuous at a point $a$, does there always exist point $b>a$ such that the function is Riemann integrable in the interval $[a,b]$?

For continuity we use the epsilon delta definition.

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  • $\begingroup$ Yes, just find a way to use the result that a continuous function is Riemann integrable on a closed interval. $\endgroup$ – Clayton Apr 11 '17 at 8:06
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Nope, this is not true. Consider the function $$ f(x) = \begin{cases} x & \quad \text{if } x \in \mathbb{Q}\\ -x & \quad \text{if } x \in \mathbb{R}\backslash\mathbb{Q} \\ \end{cases} $$

This is certainly continuous at $0$ but does not satisfy your integrability condition.

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