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I have functions continuous functions $f$, $g$ and $h$ on a bounded closed interval. I have an equation $|f - h + h - g| \leq |f - h| + |h - g|$. Can I integrate both sides of this equation over the interval?

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Note that it is in general incorrect to differentiate both sides of an inequality. –  akkkk Sep 24 '12 at 8:54
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up vote 3 down vote accepted

Any reasonable integration theory states that $f \geq 0$ implies $\int_a^b f \geq 0$. Hence inequalities like $f \leq g$ can be integrated and yield $\int_a^b f \leq \int_a^b g$.

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