If a function is continuous at point $a$, does there always exist point $b$ such that the function is Riemann integrable $[a,b]$?

Question

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.

Answer 1

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.