Let f:[a,b]->[a,b] be a continuous function. Prove that f has a fixed point, i.e. that there is a c in [a,b] such that f(c)=c

Elsa Brewer 2022-07-23 Answered
(d) Let f : [ a , b ] [ a , b ] be a continuous function. Prove that f has a fixed point, i.e. that there is a с [ a , b ] such that f ( c ) = c.
In the solution, it says that f ( a ) a and f ( b ) b but it do not seem obvious for me. If I am just given that a f : [ a , b ] [ a , b ], how do I know is this function increasing, decreasing or just a horizontal line?
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Answers (1)

kartonaun
Answered 2022-07-24 Author has 14 answers
Recall what f : [ a , b ] [ a , b ] means. It means that given any x [ a , b ], we know f ( x ) [ a , b ]. What does it mean for f ( x ) to be in the interval [ a , b ]? That a f ( x ) b. For a, a f ( a ) b and for b, a f ( b ) b. See that nothing about the shape or behavior of f is requiblack: just that the range, or image of f is a subset of [ a , b ].
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