# 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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