- #1
natasha d
- 19
- 0
Homework Statement
f:[0,1]→R where f(x)= 0 if x=0 and f(x)=1/n when 1/(1+n) < x ≤ 1/n, n [itex]\in[/itex] N.
is f Riemann integrable
Homework Equations
R integrable only when L(f) =U(f)
L(f) = largest element of the set of lower sums for n partitions
U(f) = least element of the set of upper sums for n partitions
f is R integrable if the function has finite discontinuities
The Attempt at a Solution
tried plotting a graph, where the function is constant on intervals [1/(1+n) , 1/n] since all x in between take the value 1/n
that gives me intervals on which f is R integrable except at 1/(1+n)
but that also gives ∞ such R integrable intervals, which means ∞ discontinuities...
On an interval the lower sum = the upper sum
= length of the interval X F(x)
= [(1/n)-(1/(1+n))] X (1/n)
the total integral will be the summation with n → ∞ ?