Bounded Quotient of Lipschitz functions should be continuou.

Take $f,g \in C([0,1]) $ Lipschitz, ${{f}\over{g}} \in C((0,1))$ and $\exists \infty >C>0 \forall x \in (0,1) |{{f(x)}\over{g(x)}}| < C $.

I want to prove that ${f \over g} \in C([0,1])$ is well defined and continuous on the closed intervall.

I am not a 100% sure if this is true.There are classical counter examples if either f or g is not Lipschitz. However I feel like the statement above should be true, since Lipschitz + boundedness should provide enough regularity.