# Monotonicity of the riemann integral

## Main Question or Discussion Point

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.

## Answers and Replies

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

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

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

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.