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 .