Continuity of function defined on $C_c(\mathbb{R})$ Let $X=C_c(\mathbb{R})$, the space of functions with compact support, normed with the sup norm. Define $ f: (X, \vert \vert.\vert \vert_{\infty} )\rightarrow \mathbb{R}$ as $  f= \int_{- \infty}^{\infty} x(t)dt \>\>\> \forall x \in X$
Then $f$ is continuous(T/F).
I think function is not cts but I can't construct a counterexample.

Take $x_n \in C_c(\mathbb{R})$ such that 
$$x_n(t) = \begin{cases} \frac{1}{n} & x \in [-n,n] \\ -\frac{1}{n}(x-n+1) &x \in (n, n+1] \\ - \frac{1}{n}(x+n-1) & x \in [-n-1, -n) \\ 0 & \text{otherwise}.\end{cases}$$
Then $x_n \to 0$ in $\| \cdot \|_\infty$ (that is $x_n$ tends to $0$ uniformly), but 
$$f(x_n) >\frac{1}{n}\cdot 2n = 2$$
for all $n$. Thus $f$ is not sequentially continuous (we have a sequence $x_n \to 0$ such that $f(x_n) \not\to f(0)$), which is equivalent to continuity for metric spaces.
