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