Example 7.3.1(a): Oddities of Riemann Integral
Why these ads ...
What happens when you change the value of a Riemann integrable function
at a single point?
Back
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:
- 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.
That's odd: either a change at a single point should
always matter,
or it should
never matter, regardless of the changed value.