uniform continuous in $F:(C[0,1],d_1)→(c_0,d_2)$

Question

Let $f:[0,1]\to\Bbb R$ be a countinous fuction, $$d_1(f,g)=\sup|f(x)-g(x)|$$ and $$d_2(x,y)=\sup|x_n-y_n|$$ and $$c_0 = \{ (x_n) \subseteq \mathbb{R} :\lim \;x_n = 0)\}$$

Now for $f\in C[0,1]$, we defined $F(f)=\{x_n(f)\}$ with $$x_n(f)=\int_0^{1/n} f(x) \,dx$$ and we want to prove that F is uniformly conyinous. How can I prove it?

Answer 1

Well if $d_1(f,g) < \delta$, then $|x_n(f) - x_n(g)| = |\int_0^{1/n} f-g(x)| \leq \int_0^{1/n} |f-g|(x) < \int_0^{1/n} \delta = \delta/n \leq \delta$. Hence by taking $\delta = \epsilon$, you get uniform continuity.