Example 7.3.1(a): Oddities of Riemann Integral
What happens when you change the value of a Riemann integrable function at a single point?It depends. For a function to be Riemann integrable it must be bounded. If the function was unbounded even at a single point in an interval [a, b] it is not Riemann integrable (because the sup or inf over the subinterval that includes the unbounded value is infinite). Therefore:
That's odd: either a change at a single point should always matter, or it should never matter, regardless of the changed value.
- If we change the value of a Riemann integrable function to another bounded value at a single point, the Riemann integral would not change at all (prove it).
- If we change a the value of a Riemann integrable function to infinity at a single point, then the function is no longer Riemann integrable.