The Riemann Integral and When a Function Is Integrable
Define the integral by upper and lower sums and call a function integrable when the two meet. Continuous functions qualify; a function that is discontinuous everywhere need not.
What a learner can do afterwards
- Compute upper and lower sums for a partition and compare them
- State the criterion for integrability in terms of refinements
- Give a bounded function that is not Riemann integrable
1 · Read
Chop [a, b] into pieces and trap the area between under and over estimates. Lower sums use infima per piece and upper sums use suprema. For f(x) equal to x on halves of [0, 1], the lower sum is 0.25 and the upper sum is 0.75, already pinning the true 0.5. For x squared on the same partition, the trap is 0.125 below and 0.625 above.
Refinement can only raise the floor and lower the ceiling: it raises lower sums and lowers upper sums. The integrability criterion says upper and lower sums can be made arbitrarily close by refinement. When the trap closes completely, the common value earns the name integral. Split the piece where the function swings most to narrow the trap fastest.
The Dirichlet function equals 1 on rationals and 0 elsewhere, so on [0, 1] every piece has sup 1 and inf 0. Upper sums freeze at 1 and lower sums at 0 whatever the mesh, so it is not Riemann integrable. Continuous functions on [a, b] always qualify: uniform continuity squeezes the gap to nothing.
Lower sums climb, upper sums sink, integrability means they meet, and wild oscillation like Dirichlet keeps them apart.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.