# Show that f is integrable on [0,2] and calculate the integral.

1. Apr 19, 2012

### kingstrick

1. The problem statement, all variables and given/known data

Let f:[0,2] →ℝ be defined by f(x):= 1 if x ≠ 1 and f(1) :=0. Show that f is integrable on [0,2] and calculate its integral.

2. Relevant equations

3. The attempt at a solution
i am thinking that the sup{L(p,f)} and inf{U(p,f)} is 1 at every where but where x=1. And I would assume that it is 0 where x=1. So Do I have to Rieman Summations [0,1) and (1,2]. I am confused as to how to approach this when it is not continuous.

2. Apr 19, 2012

### HallsofIvy

Staff Emeritus
Well, just one Riemann sum won't do it- to show a function is integrable, you have to show that any sequence of Riemann sums, with the maximum length of an interval going to 0, converges to the same thing.

If "P" is any partition of [0, 2] we can always make a "finer" partition, P', by adding 1 as a partition point so that L(P', f) is the sum of "x< 1" and "x> 1".