220 CHAPTER 10. BROUWER FIXED POINT THEOREM Rn∗
Now from differentiability of f at x1,
y− f(x1) = f(f−1 (y)
)− f(x1) = Df(x1)
(f−1 (y)−x1
)+o(f−1 (y)−x1
)= Df(x1)
(f−1 (y)−x1
)+o(y− f(x1))
= Df(x1)(f−1 (y)− f−1 (f(x1))
)+o(y− f(x1))
Therefore,
f−1 (y)− f−1 (f(x1)) = Df(x1)−1 (y− f(x1))+o(y− f(x1))
From the definition of the derivative, this shows that Df−1 (f(x1)) = Df(x1)−1 .