Exercises
[1GF] Consider the following \(C^β\) function of 2 variables
\[ {f}(x,y)= x^ 3+y^ 4-1 \quad . \]Check that \(\{ f=0\} =\{ (x,y)ββ^ 2: f(x,y)=0\} \) is not empty; then, for each point of the plane where \(f\) vanishes, discuss whether the implicit function theorem can be applied, and therefore if the set \(\{ f=0\} \) is locally graph of a \(C^β\) function. Also study the set \(\{ f=0\} \): is it compact? How many connected components are there?
(Please note what is shown in [1H3]).
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