Hello everyone. I'm reading Weinberg's 'Gravitation and Cosmology' and I'm having some problems understanding the definition of a 'form-invariat function'. He says:

If the previous condition was true doesn't this simply mean that ##g_{\mu\nu}^\prime## is the same function as ##g_{\mu\nu}##? Should we ask for:
\begin{equation}
g^\prime_{\mu\nu}(x^\prime)=g_{\mu\nu}(x)
\end{equation}
?

That is precisely what he means, that they are the same function.
I think that's why he used the symbol "y" instead of x's, so that it wouldn't be confused as having something to do with the coordinates x and x'.

The problem with [itex] g'_{\mu\nu}(x')=g_{\mu\nu}(x) [/itex] is that it means the two functions have same value in a single point. But because [itex] x' [/itex] associates different numbers to that point compared to [itex] x [/itex], [itex] g'_{\mu\nu}(x')=g_{\mu\nu}(x) [/itex] necessarily means [itex] g [/itex] and [itex] g' [/itex] are different functions and don't have the same form.
But [itex] g'_{\mu\nu}(y)=g_{\mu\nu}(y) [/itex] means that if we give the two functions, the same numbers, they will give equal results which means they're the same function. Note that y in the right refers to a different point than y in the left because they are same numbers in different coordinate systems.