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Prove that if functions $f : A \rightarrow B$ and $h : B \rightarrow C$ are total, then $h \circ f$ is total.

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"Total" does not normally mean "surjective". It means everywhere defined, i.e. an actual function rather than a partial function. – Chris Eagle May 4 '12 at 19:13
@Chris: I see. Well, it might not be a duplicate per se then. However if you look at the answer (or the argument) you can easily see it is really just the same thing. – Asaf Karagila May 4 '12 at 19:22
up vote 2 down vote accepted

Take any $a\in A$. Since $f$ is total, then $f(a)$ is defined, and an element of $B$, and since $h$ is total, then $h(f(a))$ is defined.

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Ah thanks! That makes sense. h is total, so, because the entire codomain of f is within B, $h(f(a))$ must be defined. – Minden Petrofsky May 4 '12 at 19:16

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