- #1
nlsherrill
- 323
- 1
Homework Statement
Let [itex]f:[0,2]-> R [/itex] be defined by[itex] f(x)= 1 [/itex] if [itex]x≠1[/itex], and [itex]f(1)= 0[/itex]. Show that [itex]f[/itex] is integrable on [itex][0,2][/itex] and calculate its integral
Homework Equations
Lower integral of f
[itex]L(f)= sup {(P;f): P \in P(I)} [/itex]
Upper integral of f
[itex]U(f)= inf {U(P;f): P \in P(I)} [/itex]
Where,
[itex]L(P;f)= \sum m_{k}(x_{k}-x_{k-1})[/itex][itex]U(P;f)= \sum M_{k}(x_{k}-x_{k-1})[/itex]And lastly, [itex]U(f)=L(f)[/itex] if the integral exists
The Attempt at a Solution
So it seems pretty obvious the integral is equal to 2, but I am not sure how to deal with this function at x=1. I tried splitting the interval up into sections, but it turns out partitions only work if the interval is closed and bounded on R, so I couldn't do any open interval stuff. Any ideas?