Is $|g'(x)|<1\ \forall x\in(a,b)$ is one of the hypothesis of the Fixed-Point Theorem?
The answer is NO. Can someone please enlightened me about this? My teacher reason is this...
Note that the condition must be $|g'(x)| \leq k < 1\ \forall x\in(a,b)$. This condition is equivalent to $g'(x)\in[-k,k]\ \forall x\in(a,b)$. The condition $|g'(x)|<1\ \forall x\in(a,b)$ is equivalent to $g'(x)\in(-1,1)\ \forall x\in(a,b)$. Observe that the two conditions are not the same.