- #1
nickolas2730
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Homework Statement
A function f :[a, b] → [a, b] is said to be a contraction on [a, b] if there exists a constant k. (0, 1) such that |f(y) - f(x)| <k|y - x| for all x, y in [a, b]. Let f be a contraction. Show that f is uniformly continuous on [a, b].
Let f : [a, b] → [a, b] be a contraction. Since it is a continuous
function by one of the previous exercises f has at least one fixed point. Prove that the fixed
point of a contraction is unique.
thank you so much
Homework Equations
The Attempt at a Solution
i tried the first question as following:
Let [tex]\epsilon[/tex]>0 and set [tex]\delta[/tex]=[tex]\epsilon[/tex]/k,│y-x│< [tex]\delta[/tex] , which implies
│f(x)-f(y)│<k*│y-x│<k*[tex]\delta[/tex]=]=[tex]\epsilon[/tex]
but i have no idea with the second question, since i couldn't even find the word contration in my entire textbook
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