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.

  • $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.