continuous sobolev function having fixed value at $0,$ being arbitrary small in norm

Question

Let $I=(0,r) $ for some $r>0$ and $W^{1,2}(I) $ the classical sobolev space of order $1$ with $p=2.$

My question is:

Let $\varepsilon>0 $ be arbitrary. Is there for any $b\in\mathbb{R}$ a continuous function $f_b\in W^{1,2}(I) $ such that

$||f_b||_{1,2}=\sqrt{||f_b||_2^2+||f_b'||_2^2}\leq\varepsilon$ and $f_b$ can be continuously extended to $0$ with $f_b(0)=b$?

I would really appreciate any help. Thats my first question, I hope I didnt do anything wrong.

Best regards

Edit: Thanks for helping

Answer 1

Since the embedding $W^{1,2}(I) \hookrightarrow C(\bar I)$ is bounded, there is a constant $c>0$ such that $$ |f(0)|\le c \|f\|_{1,2} \quad \forall f\in W^{1,2}(I). $$ This shows that for fixed $\epsilon$, the value of $|f(0)|$ cannot be arbitrarily large for $\|f\|_{1,2}\le \epsilon$.