Counterexamples needed for integration question

[Rosey24]

Homework Statement

The original question required me to show that for f(x) >= 0 for all x, f continuous, where the integral (from a to b) of f =0, that f(x) = 0 for all x in [a,b]. I did that, using a proof by contradiction.

Second part of the question requires me to show that the two hypotheses (f(x) being >= 0 and f being continuous) were required.

Homework Equations

The Attempt at a Solution

I think counterexamples would show this, but can't figure out what would make a counterexample. Do I need to take f(x)<0 for f continuous and show that its integral can't equal zero? Similarly, take f(x)>=0 but not continuous and show that its integral also can't equal zero?

 

[morphism]

I think counterexamples would show this, but can't figure out what would make a counterexample. Do I need to take f(x)<0 for f continuous and show that its integral can't equal zero? Similarly, take f(x)>=0 but not continuous and show that its integral also can't equal zero?

Almost. You want the integral to be zero. And for the first bit, you don't really need f(x)<0 for all x, just f(x₀)<0 for some x₀ in [a,b]. (Actually if f(x)<0 for all x, it won't work, because the integral won't be zero.)

 

[Rosey24]

so would taking f(x) = x^3, which is continuous, be a suitable counterexample for the first assumption?

I can't think of a function that is always positive and isn't continuous, though.

 

[morphism]

Yup. That works provided you integrate on some interval symmetric about 0, like [-1,1].

How about

[itex]
f(x)=\begin{cases}
0 &\text{if } x \neq 0\\
1 &\text{if } x = 0
\end{cases}
[/tex]

for the second one?