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.