Q: Prove that there is no norm for to make this mapping continuous I am dealing with an exercise which is as follows:
Show that there is no norm such that the set of all the mappings $T_a$ which map every element $f\in C(\mathbb{R}, \mathbb{R})$ (where the latter is the set of all continuous functions from reals to reals) to $f(a)$ for all $a\in\mathbb{R}$ are continuous. Can I get some hint to solve this?

A: Suppose there exists such a norm $\lVert \cdot \rVert$ on $C(\mathbb{R},\mathbb{R})$. Then for each $a \in \mathbb{R}$, there is a constant $c_a > 0$ such that $$\lvert f(a) \rvert = \lvert T_a(f) \rvert \le c_a \lVert f \rVert$$ for all $f \in C(\mathbb{R},\mathbb{R})$. This is a contradiction because there exists continuous functions $f : \mathbb{R} \to \mathbb{R}$ such that the set $$\left\{ \frac{\lvert f(a) \rvert}{c_a} \mid a \in \mathbb{R} \right\}$$ is not bounded.

A: But there is such a norm. Just take the supremum norm.
Added: I am assuming that $C(\mathbb R,\mathbb R)$ is the set of bounded continuous functions.
