why are definitions for (riemann) integration given this way?

Question

I've noticed some authors give the definition for Riemann integration in terms of a function $f$ with domain $[a,b]$. A definition might read "A function $f:[a,b]\to\mathbb{R}$ is Riemann integrable if ...". But don't we just want $f$ to be defined on $[a,b]$ (that is, for $[a,b]$ to be contained in the domain of $f$)? Why are we restricting $f$ in such a way? What's the point?