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 \(f\).

×

Problem Loading...

Note Loading...

Set Loading...