Inverse function theorem

Summary

• \(f:A→B\) is an arbitrary smooth function.
• If \(f\) is bijective then \(f\) has an inverse , \(g:B→A\) .
• Even if \(f\) is not bijective , \(f\) may become bijective by restrict ing the domain \(A\) .
• Example: \(f:R→R^+\) defined by \(f\left( x \right)=x^2\)
• \(f\) is not bijective and it has no inverse.
• Restricting the domain to \(R^+\) will make it bijective with inverse \(y=\sqrt{x}\)
• \(x_0\) is an arbitrary point in \(A\) .
• Inverse function theorem: if \(f'\left( x_0 \right)≠0\) then we can always find an open interval \(\left( a,b \right)\) containing \(x_0\) such that \(f\) restricted to the domain \(\left( a,b \right)\) is injective .
• Let \(f_R\) be \(f\) with domain \(\left( a,b \right)\) . By setting the codomain of \(f_R\) equal to the range of \(f_R\) will make \(f_R\) bijective and \(f_R\) has an inverse .