Can a function f be Riemann integrable over an open interval? Or does the interval considered always need to be closed?

Question

When dealing with functions that are Riemann integrable, I've always considered the function over a closed interval i.e $$f:[a,b]\to\mathbb{R}$$ but I am just wondering if you can have functions like $$f:[a,b)\to\mathbb{R} \text{ or } f:(a,b)\to\mathbb{R}$$ that are Riemann integrable also.

Answer 1

If the function $f$ is bounded in $(a,b)$ you can define

$\hat f(x)=f(x)$ if $x\in(a,b)$ and $\hat f(a)=\hat f(b)=0$.

So, if $f$ in Riemann integrable, then $\hat f$ too because $\{a,b\}$ has zero measure. In fact $$ \int_a^b\hat f(x)dx=\int_a^b f(x)dx $$