English for "prolongement" or "Fortsetzung"?

Question

I'm sorry if that's not the right place to ask for, wikipedia failed to give me the correct word...

What's the English for a function that is defined on a larger domain than the original function and coincides with it on the original domain? Is it "extension" or "continuation" or something else?

Answer 1

Both "extension" and "continuation" are OK. I would be inclined to reserve "continuation" for the analytic continuation of analytic functions (or similar cases where the extension is somehow determined by its restriction to the original domain).

Answer 2

Yes, if $X\subset Y$ and $f:X\to Z$ and $g: Y\to Z$ such that $f(x)=g(x)$ for all $x\in X$, it is possible to call $g$ an extension of $f$, or to just say that "$g$ extends $f$."

"Continuation" connotes that the extension is somehow generated by the function and the subset, but in most cases a general function could be defined quite arbitrarily outside of $X$, so it does not seem to be a good substitute for "extension."

If you'd like to know the counterpart to the phrase "$g$ extends $f$," then we would speak of "restriction." That is, "$f$ is equal to the restriction of $g$ to the set $X$."