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# What is a comparison theorem?

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A comparison theorem is a test of whether or not a mathematical object satisfies a set of predetermined properties. In calculus, a comparison can test if an integral is convergent or divergent. It is useful when one is not concerned with the actual value of an integral.

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There are times when it’s not possible to integrate, yet important to examine, the behavior of an integral. For example, if f and g are continuous functions, such that 0 ? g (x) ? f(x) for x ? ?, and function f(x) dx converges, then function g(x) dx converges, and if g(x) dx diverges, then function f(x) dx diverges.

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