Let $a$ be a real number. a) Show that the map $\delta_a$ which associates to every $\varphi \in \mathcal{D}$ the value $\varphi(a)$ is a distribution. b) Using the sequence of functions $(\varphi_n)_{n \in \mathbb{N}^*}$ of elements of $\mathcal{D}$ defined by $$\forall t \in \mathbb{R}, \varphi_n(t) = \begin{cases} \exp\left(\frac{(t-a)^2}{(t-a+1/n)(t-a-1/n)}\right) & \text{if } t \in ]a-1/n, a+1/n[ \\ 0 & \text{otherwise} \end{cases}$$ show that $\forall f \in \mathcal{F}_{sr}, T_f \neq \delta_a$.
Let $a$ be a real number.\\
a) Show that the map $\delta_a$ which associates to every $\varphi \in \mathcal{D}$ the value $\varphi(a)$ is a distribution.\\
b) Using the sequence of functions $(\varphi_n)_{n \in \mathbb{N}^*}$ of elements of $\mathcal{D}$ defined by
$$\forall t \in \mathbb{R}, \varphi_n(t) = \begin{cases} \exp\left(\frac{(t-a)^2}{(t-a+1/n)(t-a-1/n)}\right) & \text{if } t \in ]a-1/n, a+1/n[ \\ 0 & \text{otherwise} \end{cases}$$
show that $\forall f \in \mathcal{F}_{sr}, T_f \neq \delta_a$.