Consider the Sobolev Space$W^{1,2}([-1,1])$ (so one weak derivative, $L^2$-integrable)
The linear map $ev_0:W^{1,2}([-1,1])\to \mathbb{R}$ defined by$$f\mapsto f(0)$$is well defined and continuous because Sobolev embedding theorem tells us that $W^{1,2}([-1,1])\to C^0([-1,1])$ is continuous.
It follows by Riesz representation theorem that exists a function $F \in W^{1,2}([-1,1])$ representing $ev_0$, i.e.$$f(0) = \int_{-1}^1 f(t)F(t)dt + \int_{-1}^1 \partial_t f(t) \partial_tF(t) dt $$for all $f\in W^{1,2}([-1,1]).$
What is $F$ explicitely?
My initial idea was to consider $\partial_tF(t)= \mathbb{1}_{[-1,0]}(t)$ characteristic function, so $F(t) = c_0 + t\mathbb{1}_{[-1,0]}(t) +\mathbb{1}_{[0,1]}(t)$; however this yields some integral terms that do not look nice:$$\langle f, F \rangle_{W^{1,2}} = \int_{-1}^0 f(t)t\ dt + \int_{0}^1f(t) \ dt \ +f(0) -f(-1) $$