Prove that if f is continuous on [0,1] and satisfies 0 ≤ f(x) ≤ 1 there, then f has a fixed poi

Name: Prove that if f is continuous on [0,1] and satisfies 0 ≤ f(x) ≤ 1 there, then f has a fixed point; that is, there is a number c in [0,1] such that f(c)=c. Hint: Apply the Intermediate Value Theorem to g(x)=x-f(x).
Uploaded: 2022-08-17T13:54:24-08:00
Duration: 3 min 18 s
Channel: Adi S
Description: Prove that if f is continuous on [0,1] and satisfies 0 ≤ f(x) ≤ 1 there, then f has a fixed point; that is, there is a number c in [0,1] such that f(c)=c. Hint: Apply the Intermediate Value Theorem to g(x)=x-f(x).

Question

Adi S · Accepted Answer

Step 1: Define the function ( g(x) = x - f(x) ) for ( x in [0,1] ).

Question

Prove that if $f$ is continuous on $[0,1]$ and satisfies $0 \leq f(x) \leq 1$ there, then $f$ has a fixed point; that is, there is a number $c$ in $[0,1]$ such that $f(c)=c$. Hint: Apply the Intermediate Value Theorem to $g(x)=x-f(x)$.

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