I am trying to find the root of $f(x)=\arctan(x)$ by using successive iteration. There are some conditions to apply this in successive iteration .
1) The function has to be continuous.
2) $\sup|f'(x)|<1$
3) $f(a,b)\subset (a,b)$, where $f(x)$ is continuous in the interval $(a,b)$.
When I applied it in Matlab, it went to $0$ but the problem is that $0$ is a root but $f'(0)=1$ and it is not smaller than $1$. According to fixed point theorem, it must be $|f(x)|<1$.
So can we apply $f(x)$ in fixed point theorem? Or am I wrong? Can you help me please? Thank you for your helping.