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.