One of the common examples to show that $H^1(\Omega)$ functions are not continuous in an open subset of $\mathbb{R}^n$, $n>1$ is the following.
Let $\Omega =B_1(0)$ and $$f(x,y)=|\log(\frac{1}{\sqrt{x^2+y^2}})|^k$$ for $k \in (0, \frac{1}{2})$. Then $f \in H^1$, but it's not continuous.
I can't prove that the partial are in $L^2$. I'll show here what I tried so far.
First of all, since I am in $B_1(0)$, then $x^2+y^2<1 \Longrightarrow \sqrt{x^2+y^2} <1 \Longrightarrow \frac{1}{\sqrt{x^2+y^2}}>1$, hence the argument of $\log$ is greater than $1$ and then the logarithm is positive on the unit ball, so I can remove the absolute value.
Then I compute the partial: $$\partial_x f(x,y)= \frac{k x \log \Bigl(\frac{1}{\sqrt{x^2+y^2}}\Bigr)^{k-1}}{x^2+y^2}$$
Using polar coordinates: $$\int_0^{2 \pi} \int_0^1 \frac{k^2 r^2 \cos(\theta)^2 \log(\frac{1}{r})^{2(k-1)}}{r^4} r dr d \theta = \int_0^{2\pi} \cos(\theta)^2 \int_0^1 k^2 \frac{\log(1/r)^{2(k-1)}}{r}dr$$
but I don't know how to handle the latter integrals. The first one is not $0$ of course, but I don't know if the other one is convergent or not. What should I do here?