From my exam

Calculus Level pending

Consider \(\phi \in C^1 (\mathbb{R})\).

(1): If there are two distinct fixed points of \(\phi\) then there is a point \(c \in \mathbb{R}\) that \(\phi ^{'} (c)=1\).

(2): If there is a sequence \((x_{n})\) of distinct fixed points of \(\phi\) such that \((x_{n}) \rightarrow d\), then \(\phi ^{'}(d)=1\).

Clarifications:

  • \(f \in C^1\) means that the first derivative of \(f\) is continuous.
  • A fixed point of a function \(f\) is a point, \(z\), that \(f(z)=z\).
  • \(f^{'}\) is the first derivative of \(f\).
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