i have the following homework:
Let $g : \mathbb{R} \to \mathbb{R}$ be continous and $X = C([a,b])$ with the metric $ d(x,y) := \sup_{t \in [a,b]} |x(t) - y(t)| $ show that $F: X \to \mathbb{R}$ with $ F(x) = \int_a^b g(x(t)) \mathrm{d}t $ is a continous function.
I was able to show a much weaker proposition, namely that the functional $ F'(f) = \int_a^b f(t) \mathrm{d}t $ is uniformly continous.
My proof: For $\varepsilon > 0$, set $\delta := \frac{\varepsilon}{b-a}$, if $d(f,g) < \delta$ then \begin{align*} |F'(f) - F'(g)| & = | \int_a^b f(t) \mathrm{d}t - \int_a^b g(t) \mathrm{d}t| \\ & = | \int_a^b f(t) - g(t) \mathrm{d}t | \\ & \le \int_a^b |f(t) - g(t)| \mathrm{d}t \\ & \le \int_a^b \delta \mathrm{d}t \\ & = \delta \cdot (b-a) = \varepsilon \end{align*} But i am not able to generalize my results, do you have any hints for me?