Functional defined on the space of functions with compact support

Question

Let $X=C_c(\mathbb{R})$, the space of functions with compact support, normed with the max norm. Define $\Gamma: X \rightarrow \mathbb{R}$ as: $$ \Gamma f= \int_{- \infty}^{\infty} f(t)dt \>\>\> \forall f \in X$$ Prove $\Gamma$ is not continuous.

Answer 1

Let $f$ be a continuous function such that $f(x)\in [0,1]$ for all $x$, $f(t)=1$ if $t\in [-1,1]$ and $f(t)=0$ if $|t|>2$. Then define $f_n(x):=f(nx)/n$. Since $ \int f_n(x)\mathrm dx=\int_\mathbb R f(x)\mathrm d x\gt 0$ and $\lVert f_n\rVert=1/n$, $\Gamma$ is not continuous.