Calculus question

Question

Suppose that f : [0, 1] −→ [0, 1] is a continuous function. Prove that f has a fixed point in [0, 1], i.e., there is at least one real number x ∈ [0, 1] such that f(x) = x

Answer 1

Let g(x) = f(x) - x.

 

Then since f(0) ≥ 0 and f(1) ≤ 1, we have

 

g(0) ≥ 0 and g(1) ≤ 0

 

If g(0) = 0 then x=0 is a fixed point and if g(1) = 0 then x=1 is a fixed point.

 

If neither of these is true, then by the intermediate value theorem, there is a c ∈ [0,1] such that g(c) = 0.

 

Then f(c) = g(c) + c = c so x=c is a fixed point of f(x).

 

Q.E.D.

Answer 2

f(x) is a linear function with a slope of 1.  Linear functions are continuous in all directions and the domain of a linear function is ALL real numbers.