Continuous functions are Riemann integrable

Continuous implies Riemann integrable: Let $a<b$. If $f:[a,b]\to\mathbb{R}$ is continuous (as a continuous map ), then $f$ is a Riemann integrable function on $[a,b]$.

In practice, this guarantees the existence of the Riemann integral for most elementary functions, and it interacts well with the mean value theorem for integrals and the fundamental theorem of calculus .