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I assume that, when a math book says a function is greater than the other, i.e. $f > g$ where $f$ and $g$ are functions, $f$ is greater than the $g$ in the pointwise, everywhere on the domain of these functions. So I also assume that, domain of these functions should agree. Am I right? Or what are the other implications of this statement?


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Your interpretation seems quite right (with range instead of somain - typo?), but it might depend on the context. Some author might write $f>g$ e.g. for functions defined on $\mathbb N$ if we have $f(n)>g(n)$ for almost all $n$. In other contexts, one might write $f>g$ if the domain of $g$ is a strict subset of the domain of $f$ and $g$ is the restriction of $f$ to that domain. – Hagen von Eitzen Nov 7 '12 at 19:06
Thanks for your comment and editing! – oeda Nov 7 '12 at 19:13
up vote 3 down vote accepted

If the domain of $f$ is the set $F$ and the domain of the function $g$ is $G$, then a typical interpretation of $f>g$ is: $$\text{For all $x \in F \cap G$, we have $f(x) > g(x)$.}$$ It is not necessary that the range of the functions should agree. For instance, $f(x) = 1$ has range $\{1\}$, whereas $g(x) = 1-e^{-x}$ has range $(-\infty,1)$.

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