title: Numerical Methods - Checking Domain of Phi(x)
body: <p>I have just started a MSc in Scientific Computing, and being an engineer my knowledge of real analysis is somewhat limited.</p>
  
  <p>When rewriting f(x) = 0 as a fixed point formulation Phi(x) = x, it is stressed that we must check that for x in the interval [a,b] that Phi(x) maps into the same interval.</p>
  
  <p>Is there a general real analysis method of checking this, using the Mean Value Theorem for example, or do I need to use simpler calculus method of checking the minimum and maximum values of Phi(x). The simpler calculus method doesn’t seem to be satisfactory or formal enough in a real analysis sense.</p>
  
  <p>Any assistance would be appreciated.</p>
  
  <p>Kind regards</p>
  
  <p>John</p>