If f is continuous on [a,b], it is uniformly continuous, and for any ε there is a δ such that points within δ of each other have functional values within ε/(b-a) of each other. The upper and lower sums differ by at most ε, and the limit exists.
Since integrals can be added together, piecewise monotonic or continuous functions are also integrable. Just integrate each chunk.
If a complex or vector function is continuous, the same holds for its components, hence the vector function is integrable.