Give $R^I$ the uniform metric, where $I = [0, 1]$. Let $C(I, R)$ be the subspace consisting of continuous functions. Show that $C(I, R)$ has a countable dense subset, and therefore a countable basis.
I know to show that having a countable dense subset implies having a countable basis.
I'm doing this exercise in Munkres book and got no clue about the solution. Hope someone can help me solve this.
Hint Given in the Book is...Consider those continuous functions whose graphs consist of finitely many line segments with rational end points.