From http://en.wikipedia.org/wiki/Semicomputable_function, we have:
"If a partial function is both upper and lower semicomputable it is called computable."
Is this the same kind of "computable (partial) function" as defined here? http://en.wikipedia.org/wiki/Computable_function
If so, can you provide a proof?
If not, is there some straightforward way they are related?
The semicomputable functions being described are functions from $\mathbb{Q}$ to $\mathbb{R}$. These are an entirely different kind of function than the functions from $\mathbb{N}$ to $\mathbb{N}$. So the two notions of "computability" are not the same, because they are about different kinds of objects.
They are related, of course, in that they both use Turing machines somehow. But it is very different to use a Turing machine to compute a natural number than it is to use a Turing machine to compute a real number.