It's just telling you to consider the function that maps , which happens to be continuous. That's important because if the determinant is nonzero at a, then it is nonzero in a neighborhood of a, by continuity.
This is not a problem to be solved, but I could not understand something important from the text. Here it is.
Let be positive integers, and let be continuous real-valued functions on an open subset of that contains the point with . Suppose that for each
exists and is continuous on the given open subset and that the determinant
is not zero.
Then there exists open subsets with and such that there is a unique function such that for each
....
There is some line written such that "the continuous function on whose value at is det is not zero at
Question
1. Why does such a continuous function on whose value at equals det ?
I have no idea why a continuous function has value equals to det of something ?
2. It says "whose value at and "not zero at ". What is this? Why it mentioned two points , and ?
If one could explain without using linear algebra, I would be thankful.
Thank you very much.