Question 3. Let (X,p) be a compact metric space. Suppose that f: X→ X is a function satisfying that for all r, y e X with a y, P(f(x), f(y)) < p(x, y). Define g: XR as g(x):= p(x, f(x)). (a) Show that g is continuous on X. (Hint: Use the usual e-6 definition of continuity, and use triangle inequality to show that g(a) s p(x,y)+ g(y) + p(f(x), f(y)).) (b) Show that there is a e X such that g(a) = infrex {g(x)}. (c) , Show that f has a unique fixed point. (Hint: f may not be a contraction, so don't try using Banach's fixed point theorem. Show instead that g(a) = 0.)

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