A Continuous Function on a Closed Interval is Riemann Integrable

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $a,b\in\mathbb{R}$ with $a<b$, and let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$. Integrability on $[a,b]$ and on a closed subinterval of $[a,b]$ are as in The Riemann Integral on a Closed Interval via Lower and Upper Integrals (iv) and The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vi).

(i) $f$ is bounded, that is, there is $B\in\mathbb{R}$ with $|f(t)|\le B$ for every $t\in[a,b]$, and $f$ is integrable on $[a,b]$.

(ii) For all $u,v\in[a,b]$ with $u<v$, $f$ is integrable on $[u,v]$.