If we let $\Omega\subset\mathbb{R}^d$ with $d=1,2,3$ and define $\mathcal{H}^1(\Omega)=(w\in L\_2(\Omega): \frac{\partial w}{\partial x\_i}\in L\_2(\Omega), i=1,...,d)$. My tutor has repeated several times:


1. If $d=1$ then $\mathcal{H}^1(\Omega)\subset\mathcal{C}^0(\Omega)$.
2. If $d=2$ then $\mathcal{H}^2(\Omega)\subset\mathcal{C}^0(\Omega)$ but $\mathcal{H}^1(\Omega)\not\subset\mathcal{C}^0(\Omega)$.
3. If $d=3$ then $\mathcal{H}^3(\Omega)\subset\mathcal{C}^0(\Omega)$ but $\mathcal{H}^2(\Omega)\not\subset\mathcal{C}^0(\Omega)$.


I was interested in trying to show these relationships. Does anyone know any references that would be useful. 


Thanks in advance. 



When is Sobolev space a subset of the continuous functions?
I understand from your post that you'd like to show those facts by yourself first, and not necessarily to approach the whole theory now (I like your approach). Trivial hint: start with smooth functions with compact support in $\Omega$, and try to bound their $L^\infty$ norm in terms of the $H^d$ norm. Also, I suggest that you try building counter-examples by yourself for the case of non-inclusions.
Reference: Brezis' book of Functional Analysis may give you nice hints. 


