Suppose f(x) is continuous on the interval [0,1] with 0< = f(x)< = 1 for all x. prove that f has a fixed point; that is, there exists a c in the interval [0,1] such that f(c) = c. (hint: you need to use the intermediate value theorem on the function h(x) = f(x) – x.)

The equation    is equivalent to    (subtract x from both sides).if we let  , then the solution(s) to this equation is equivalent to finding the fixed points of  :     is continuous because the difference of two continuous functions is still continuous (the second function can be viewed as y=x, which is just a linear function).let us take the given range for  :     if we subtract  from everything, then we will get the possible  range of values for  at a point since  . we can substitute in h(x) at the last step.    for  , we have    this tells us the value of  is between 0 and 1 inclusive.for  we have    this tells us the value of  is between -1 and 0 inclusive.from these results, we can conclude thath(0) can be above the x-axis or equal to 0.h(1) can be below the x-axis or equal to 0.therefore, 0 is an intermediate value between  and  .  is a continuous function; it has to attain a value of 0 somewhere in the interval of  .by the intermediate value theorem, there is a number  in the interval  that satisfies the equation  .and we established before that  is the same as  .