Checking continuity of a functional

Question

Let $p \in \mathbb{C}[x]$ and $\mathbb{C}[x]$ is endowed with the norm $\|p\| = sup \{|p(t)|: t \in [0,1]\}$. I need to check if the functional $f(p) = p'(0)$ is continuous or not. I tend to think it's not. To prove it I tried to find some sequence in the unit ball which goes to unbounded set but I couldn't. I used the most simple ones like $2^{-n}(x+1)^n$ but it's image is bounded. What else can I do?

Answer 1

Try $p_n(x)=(x-1)^n$. Then $||p_n||=1$ and $|f(p_n)|=|p_n'(0)|=n=n ||p_n||$