Continuity of functional in $E\subset C^1([0,1])$

Question

Let $$E:=\{x\in C^1([0,1]):x(0)=x(1)=0\}.$$ Then $(E,\|\cdot\|)$ is a normed space when $$\|x\|=\sup_{t\in[0,1]}\{|x'(t)|\}.$$ Now fix $s\in[0,1]$ and set $f_s:E\rightarrow\mathbb R$ as $f_s(x)=x(s)$. Clearly, $f_s$ is a linear functional. I'd like to prove $f_s$ is continuous. But I couldn't do that using only the definition and searching for a good $K$. If someone could give me a hint, I'd be grateful. Thanks in advance!

Answer 1

$|f_s(x)|=|x(0)+\int_0^{s} x'(t)\, dt| \leq\|x\|$ (because $x(0)=0$). If $x,y \in E%+$ then $|f_s(x)-f_s(y)|=|f_s(x-y)| \leq \|x-y\| <\epsilon $ whenever $\|x-y\|<\epsilon$.