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chrischoi614
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Homework Statement
comparing two partitions: Let f be defined on the interval [0,2] by f(x)=0, if 0≤x≤1
and f(x)=1 otherwise. Consider two partitions P={0,1,2} and
Q={0,1-[itex]\epsilon[/itex], 1+[itex]\epsilon[/itex],2}, and calculate the difference of the upper and the lower sum with respect to each partition. By using partitions like Q prove directly from the definition of integrability that the function f is integrable and calculate the integral.
Homework Equations
spf=ƩJ1mj(xj-xj-1)
and replace m with M for Spf
the definition of integrability I use in my textbook is:
If f is a bounded function on [a,b], the following conditions are equivalent:
a. f is integrable on [a,b]
b. For every ε>0 there is a partition P of [a,b] such that SPf-sPf<ε
The Attempt at a Solution
I have made an attempt on solving for the upper and lower sum of P and I got 1+1+0 since the points {1,2} in P is 1 and {0} is 0. For Q I got the upper and lower sum as 2 as well as {0,1-ε} is in 0 and {1+ε,2} is in 1. However I don't know if I'm right or not. Neither do I know how to proceed with this question. Thank you very much in advance for anyone that's going to help me!