A sequence in $C([-1,1])$ and $C^1([-1,1])$ with star-weak convergence w.r.t. to one space, but not the other The functionals
$$
  \phi_n(x) = \int_{\frac{1}{n} \le |t| \le 1} \frac{x(t)}{t} \mathrm{d} t
$$
define a sequence of functionls in $C([-1,1])$ and $C^1([-1,1])$.
a) Show that $(\phi_n)$ converges *-weakly in $C^1([-1,1])'$.
b) Does $(\phi_n)$ converges *-weakly in $C([-1,1])'$?
For me the limit functional
$$
 \int_{0 \le |t| \le 1} \frac{x(t)}{t} \mathrm{d} t
$$
is not well defined so i have trouble evaluating the condition of convergence? Do you have any hints?

@Davide Giraduo
Regarding your derivation, i came to another result:
\begin{align*}
 \int_{1/n \le |t| \le 1} \frac{x(t)}{t} \mathrm{d} t & = \int_{-1}^{-1/n} \frac{x(t)}{t} \mathrm{d} t +  \int_{1/n}^1 \frac{x(t)}{t} \mathrm{d} t \\ 
                                                                                 & =  - \int_{1}^{1/n} \frac{x(-t)}{t} \mathrm{d} t +  \int_{1/n}^1 \frac{x(t)}{t} \mathrm{d} t \\ 
   & =  \int_{1/n}^{1} \frac{x(-t)}{t} \mathrm{d} t +  \int_{1/n}^1 \frac{x(t)}{t} \mathrm{d} t \\
                      & = \int_{1/n}^{1} \frac{x(-t) + x(t)}{t} \, \mathrm{d} t
\end{align*}
Which has others sign's?
