TITLE: Prove that $A= \left\lbrace f \in X: \int_0^1f(x) dx>1 \right\rbrace$ is an open set in $(X,d)$.
QUESTION [5 upvotes]: Let $X=C[0,1]$ be a set of real continuous functions on $[0,1]$. Given these metrics below 
$$d(f,g)=\max_{x \in [0,1]} e^{-x^2} \vert f(x)-g(x) \vert\ , \ \forall f,g \in X$$
$$p(f,g)= \int\limits_0^1{\vert f(x)-g(x)\vert dx}, \ \forall f,g \in X$$.
Question 1: I have already proved that $p(f,g) \le e \cdot d(f,g) $, so that with $f \in X$ and for all sequence $(f_n) \subset X$, I have if $\lim d(f_n,f)=0$ then $\lim p(f_n,f)=0$. Moreover, if $\lim p(f_n,f)=0$ then $\lim d(f_n,f)=0$ does not happen. However I can't find the example to point out it doesn't happen. 
Is there any example ? And how can you find out this example ?
Question 2: Prove that $A= \left\lbrace f \in X: \int\limits_0^1f(x) dx>1 \right\rbrace$ is an open set in $(X,d)$.
For this question I intend to write it as an inversion of open set and the mapping is continuos but I stuck. Do you have any solution for this ? Thank you

REPLY [2 votes]: Let $f_n(x)=x^n$ and $f(x)=0$. It is easy to see
$$ \int_0^1|f_n(x)-f(x)|dx=\frac1{n+1}\to 0 $$
as $n\to\infty$. However
$$ d(f_n,f)=\max_{x\in[0,1]}x^ne^{-x^2}=\left(\frac{n}{2e}\right)^{n/2}\to\infty $$
as $n\to\infty$.