Question

2, Let f : U → Rm be Ci and let B be a compact connected subset of U Show that there exists a constant M such that for all a, y e B. (Hint: use the mean value theorem). Find an example which shows that the assumption that B was compact is essential

B is a connected ball of finite radius

Answer 1

Since
IR
is $C^1$ so we can apply the mean value theorem on any ball .

So there exists such that

$||f(x)-f(y)||=||f^{'}(z)||||x-y||$$\forall x,y\in B$.

Now it is given that the space is compact and we know that any continuous function on a compact set is bounded.

Since $f^{'}$is continuous and hence bounded we can find such that $||f^{'}(z)||\le M \forall z\in B$ .

Thus we have

$||f(x)-f(y)||=||f^{'}(z)||||x-y||$

$\implies ||f(x)-f(y)||\le M||x-y||$

$\forall x,y\in B$.