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.

Assume that $\|f\|\leq M$ for some $M>0$, then for all $x\in C_{c}({\bf{R}})$ with $\|x\|_{\infty}\leq 1$, one has $|f(x)|\leq M$. Now find some $x\in C_{c}({\bf{R}})$ such that $0\leq x\leq 1$ and that $x=1$ on $[-M,M]$, then $|f(x)|\geq\displaystyle\int_{-M}^{M}x(t)dt=2M$ but $\|x\|_{\infty}=1$.
