Continuity, Uniform Continuity

Let $\displaystyle g: \mathbb{R} \rightarrow \mathbb{R}$, be continuous. If $\displaystyle F: C([0,1], \mathbb{R}) \rightarrow C([0,1], \mathbb{R})$ by $\displaystyle F(f) = g \circ f$, show that $\displaystyle F$ must also be continuous. Likewise, if $\displaystyle g$ is uniformly continuous, then show $\displaystyle F$ is also uniformly continuous.

I would like to show the result directly, using definitions and basic results, but I can't seem to keep my work straight enough. What do you think--should I try it again, or is there an alternate approach that might make my life simpler here?