- #1
theneedtoknow
- 176
- 0
Homework Statement
PRove that, if f is continuous on [a,b] and the integral from a to b of |f(x)|dx = 0 (thats absolute value of f(x) ), then f(x) = 0 for all x in [a,b]
Homework Equations
The Attempt at a Solution
HEre is my attempt...its not very rigorous which is why I'm asking for any tips that could make this a more formal proof:
Proof by contradiction:
Assume f(x) is not equal to zero for all x in [a,b]
then there exists an x1 element of [a,b] such that f(x1) > 0
Then by properties of integrals we can integrate f(x) from a to x1 and from x1 to b and add them together
since f(x1) is greater than zero and not just a removable discontinuity (since f is stated as continusous), the area bound by x = a, x = x1, y = 0 and y = f(x) must also be greater than zero and therefore the integral from a to x1 of |f(x)|dx is also greater than zero. Since the function |f(x)| is always nonnegative, the integral from x1 to b of |f(x)|dx is also nonnegative. Since the integral from a to x1 is positive and from x1 to b is non-negative, then their sum must also be positive. But this congradicts the assumption that the integral from a to b of |f(x)|dx = 0 for all x in [a,b] , so f(x) must be 0 for all x in [a,b]