Riemann criterion
Checking integrability straight from the definition means computing two suprema/infima over all partitions, which is awkward. The Riemann criterion replaces that with a single, checkable test: a function is integrable exactly when you can find one partition whose upper and lower staircases are as close as you please. You only need to exhibit a good partition, not optimize over all of them.
Precisely, a bounded f on [a, b] is Riemann integrable if and only if for every epsilon > 0 there exists a partition P with U(f, P) - L(f, P) < epsilon. The difference U(f, P) - L(f, P) equals sum (M_k - m_k)(x_k - x_(k-1)), the total area of the gap between the two staircases; making it small means the over- and under-estimates pinch together.
This is the integration analogue of the Cauchy criterion for sequences: it certifies the existence of a limit without naming it. One direction is easy from monotonicity of sums under refinement; the other uses that the gap controls the difference of upper and lower integrals. It is the workhorse used to prove that continuous and monotone functions are integrable, by choosing partitions fine enough to force each M_k - m_k small.
If f is continuous on [a, b] it is uniformly continuous, so given epsilon there is delta with |f(x) - f(y)| < epsilon/(b-a) whenever |x - y| < delta; any partition of mesh < delta then has each M_k - m_k < epsilon/(b-a), giving U - L < epsilon. Hence every continuous function is integrable.
Uniform continuity plus the Riemann criterion is the cleanest proof that continuous functions are integrable.