Use the intermediate value theorem to prove that if f : [ 0 , 1 ] → [ 0 , 1 ] is continuous, then there exists c ∈ [ 0 , 1 ] such that f ( c ) = c Suppose that f ( 0 ) &lt; f ( 1 ). Consider now, f ( 0 ) &lt; c &lt; f ( 1 ). By the intermediate value theorem, ∃ b ∈ [ 0 , 1 ] : f ( b ) = c .Now we need to show that its possible that b = c, how?

Question

Use the intermediate value theorem to prove that if   f  :  [  0  ,  1  ]  →  [  0  ,  1  ] is continuous, then there exists   c  ∈  [  0  ,  1  ] such that   f  (  c  )  =      c  Suppose that   f  (  0  )  &amp;lt;  f  (  1  ). Consider now,   f  (  0  )  &amp;lt;      c    &amp;lt;  f  (  1  ). By the intermediate value theorem,   ∃  b  ∈  [  0  ,  1  ]  :  f  (  b  )  =      c  .Now we need to show that its possible that   b  =  c, how?

madleeinluvec · Accepted Answer

Write   g  (  x  )  =  f  (  x  )  −      x  ,   g  (  0  )  ≤  0  ,  g  (  1  )  ≥  0, so there exists   c such that   g  (  c  )  =  f  (  c  )  −      c    =  0.

Suppose that f(0)<f(1). Consider now, f(0)<sqrt(c)<f(1). By the intermediate value theorem, EEb in [0,1]:f(b)=sqrt(c). Now we need to show that its possible that b=c, but this is exactly where I am stuck.

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