f(f(c))=c.md

Suppose f : R → R is continuous and that there is a point c such that f(f(c))=c. Show that f has a fixed point.

---

Let g(x) = f(x) - x. Note that g is continuous. Consider two points on g:

• g(c) = f(c) - c

• g(f(c)) = f(f(c)) - f(c) = c - f(c)

g(c) = -g(f(c))

If c = f(c), then c is a fixed point of f. Otherwise if c ≠ f(c), the intermediate value theorem implies ∃ a : g(a) = 0.

(0 = g(a) = f(a) - a)) ⇒ (f(a) = a) ⇒ a is a fixed point of f.