Exercise 9, § 1

Let $f$ be a regulated function on a compact interval $I = [a,\ b]$. Show that for every $\varepsilon > 0$ there exists a continuous function $g$ on $I$ such that $\int_a^b \|f(t) - g(t)\| dt \leq \varepsilon$ (reduce to the case where $f$ is a step function). Deduce that there exists a polynomial $h$ (with coefficients in $E$) such that $\int_a^b \|f(t) - h(t)\| dt \leq \varepsilon$.