39f must be bijective
and every bijective continuous function from [0,1] to [0,1] is monotonic
f is a continuous function that maps the closed unit interval I = [0,1] into itself. Prove that if f(f(x)) = x for all x ε I, then f is monotonic
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5 Answers
39
341ya. but we only need it to be injective and continuous, and for that note that if f(x1) = f(x2) then f(f(x1)) = f(f(x2)) which implies x1=x2
Which theorem allows us to relate this fact to f being monotonic?
3we have f'(f(x))f'(x)=1
and f'(f(x))=\lim_{h\rightarrow 0}\frac{f(f(x+h))-f(f(x))}{h}
and f(f(x))=x for all values x in the given interval so f'(f(x))=1
so from the first eqn we have f'(x)=1
66The last result in #3 easily follows from intermediate value property.