# Let the function be defined on the unit hypercube

Oftentimes I see in the derivation of an algorithm or in a mathematical proof the phrase that the function under consideration is assumed to be defined on the unit hypercube $[0, 1]^n$, which is claimed to be imposing practically no loss of generality. I am curious to know what are the actual assumptions we are making here about the function. Should the function, for instance, be continuous on its domain? Or may be the domain should have no holes? I would be grateful if somebody could summarize the assumptions one needs in this context and show that, indeed, the problem can be reduced to $[0, 1]^n$.

Thank you.

Regards, Ivan

## 1 Answer

I think, that both in this particular case and in general case, when an author claims that without loss of generality we may assume something it means not a reference to some general facts but that it (for instnce, such a reduction) is obvious for everyone who understand the investigated situation or problem. :-)

• I’ve renamed the post and slightly changed the description to draw less attention to the phrase “without loss of generality.” I suppose there is some loss, but it’s negligible for practical applications. – Ivan Jan 12 '15 at 8:51