There exist a function $u(x)\in C([0,1]),~ u(0)=u(1)=0$ such that $u \notin H_{0}^{1}((0,1))$?

Question

There exist a function $u(x)\in C([0,1]),~ u(0)=u(1)=0$ such that $u \notin H_{0}^{1}((0,1))$? Justify your answer!

Thanks.

Answer 1

Try checking if all of the properties of $H^1_0((0,1))$ are met for the following function: $$ f(x) = x\sin\frac{\pi}{x} $$

Intuition: construct an example of a continuous function, it oscillates so much that its derivative may not be $L^2$-integrable.