Question

Verify that if f(x) is continuous on [a,b], then the absolute value of the integral from a to b of f(x) is less than infinity.

| ∫[a,b] f(x)dx | < ∞

Answer 1

Are you familiar with the extreme value theorem?

Answer 2

Since you can show that the image of the set is bounded above, it is simple to show that the upper/lower sums for the riemann definition of the integral are also bounded above.