Finding bounded linear functionals on $L^{\infty}(\mathbb{R})$

Question

I'm trying to prove the next two propositions:

a) There is a nonzero bounded linear functional on $L^{\infty}(\mathbb{R})$ which vanishes on $C(\mathbb{R}).$

b) There is a bounded linear functional $\lambda$ on $L^{\infty}(\mathbb{R})$ such that $\lambda(f)=f(0)$ for each $f\in C(\mathbb{R}),$

where $C(\mathbb{R})$ is the space of all bounded real valued continuous functions on $\mathbb{R}$ equipped with the sup norm .

I'm stuck proving this. I guess Hahn-Banach could solve this:

Part b) with $g:C(\mathbb{R})\rightarrow\mathbb{C}$ defined by $g(f)=f(0).$ Such map is linear and I think is bounded. If this were the case, a Corollary of Hahn-Banach solves this.

With part a) I don't get any useful.

How to prove this?

Answer 1

For (a) you could take the space of $L^\infty$ functions that have (a representive with) left and right limits at zero, define $L(f)$ to be the right limit minus the left limit on this space, and extend by Hahn-Banach. This will be nonzero on the Heaviside step function.