(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?

Question

(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?

kartonaun · Accepted Answer

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  ].

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

Answered question

Answer & Explanation

New Questions in Calculus 2