Answer
It is possible that $f(x)/g(x)$ be discontinuous at a point of $[0,1]$.
Work Step by Step
It is possible that $f(x)/g(x)$ be discontinuous at a point of $[0,1]$ even when $f(x)$ and $g(x)$ are both continuous in this interval. This will happen when $g(x)=0$ at a point of $[0,1]$.
Recall the properties of continuous functions: $f$ and $g$ are continuous at $x=c$, but $f/g$ is only continuous at $x=c$ if $g(c)\ne0$. This is because if $g(c)=0$, then $f(c)/g(c)=f(c)/0$, which is not defined, meaning that $f/g$ is discontinuous at $x=c$.
This applies to our situation here. If $g(x)=0$ at any single point $x=c\in[0,1]$, then $f(c)/g(c)$ would be discontinuous at $x=c$, even though $f(x)$ and $g(x)$ are continuous there.
