Answer
See the explanation below.
Work Step by Step
Yes.
Theorem 7.5 states that, if the functions $f$ and $g$ are continuous at $x=c$, then $f/g$, provided $g(c)\neq 0$, is continuous at $x=c.$
Note the "provided $g(c)\neq 0$."
So, let f be any polynomial, say $f(x)=x+1$ and let $g(x)=10x-1$
Both $f$ and $g$ are continuous on $[0,1]$ but $( f/g)(x)=\displaystyle \frac{x+1}{10x-1}$ is not defined for $x=0.1$, meaning that it is discontinuous at $x=0.1\in[0,1]$.
