In fixed point theorem, If g is a continuous function $ g(x) \in [a,b]$ for all $ x \in [a,b]$, then g has a fixed point in $[a,b]$ i.e. $ c \in [a,b]$ such that $g(c)=c$
According to this theorem, We have exactly one fixed point or at least one fixed point?
Without loss of generality you can reparametrize $[a,b]$ with $[0,1]$.
Given $g$ define $f(t)=g(t)-t$.
$f$ is continuous. Since $g$ takes value in $[0,1]$, then and $f(0)\geq 0$ and $f(1)\leq 0$. By the Intermediate Value Theorem, there is at least one $t\in[0,1]$ so that $f(t)=0$, which implies $g(t)=t$.