Monotonicity of the riemann integral

[Markjdb]

Hi everyone,

For integrable f,g:\left[a,b\right]\rightarrow\mathbb{R} with f(x)\leq g(x) for all x\in\left[a,b\right], it's a basic property of the riemann integral that
\[\int_a^b f(x)\,dx \leq \int_a^b g(x)\,dx\]

My question is whether the strict version of this inequality holds, i.e. if we have the same hypotheses as above, except with f(x)<g(x) for all x\in\left[a,b\right], then do we get the following inequality?
\[\int_a^b f(x)\,dx < \int_a^b g(x)\,dx\]

This question arose while trying to solve a rather different problem; I feel like it's not true in general, but I haven't yet come up with a counterexample.

 

[Preno]

Yes, you get the strict inequality. The integral of a positive function is obviously positive.

 

[JG89]

If you take a = b then the strict inequality is not true.

 

[sutupidmath]

Like jg89 pointed out, it holds as long as the lower and the upper limits of integration are not the same.

 

[adriank]

Yes, you get the strict inequality. The integral of a positive function is obviously positive.

Is this obvious?

A Riemann-integrable function on [a, b] (with a < b) is continuous almost everywhere, so in particular it's continuous at one point; if this ensures that if the function is strictly positive, then its integral is positive.