(10 X 3 = 30 points) Consider the Brouwer's Fixed Point Theorem

(a) Write it clearly in the "If A, then B" format. Draw a rough picture and label it appropriately to explain exactly what the theorem tells us.

(b) Provide an example to explain what the theorem says. Explain why the example you have provided satisfies all the conditions that are required by the theorem.

(c) Construct three examples to explain why the function may not have a fixed point

(a) if the function is not continuous,

(b) if the input set is not convex, or

(c) if the input set is not compact.