- #1
Unassuming
- 167
- 0
Homework Statement
Suppose f is continuous on [a,b] with f(x) >= 0 for all x in [a,b].
If [tex] \int_a^b \! f = 0 [/tex] , prove that f(x)=0 for all x in [a,b].
Homework Equations
The Attempt at a Solution
Assume [tex] \int_a^b \! f = 0 [/tex].
Then the lower and upper sums equal 0 also. I looked at the definitions of these and came up with,
[tex] \sum_{i=1}^n ( inf\{ f(x) : x_{i-1} \leq x \leq x_i \} )( \bigtriangleup x_i )=0 [/tex].
This looks obvious to me but that usually means I'm missing something because I don't know where to use that f is continuous.