differential equations - When is Sobolev space a subset of the continuous functions?

If we let $Omegasubsetmathbb{R}^d$ with $d=1,2,3$ and define $mathcal{H}^1(Omega)=(win L_2(Omega): frac{partial w}{partial x_i}in L_2(Omega), i=1,...,d)$. My tutor has repeated several times:

1. If $d=1$ then $mathcal{H}^1(Omega)subsetmathcal{C}^0(Omega)$.


2. If $d=2$ then $mathcal{H}^2(Omega)subsetmathcal{C}^0(Omega)$ but $mathcal{H}^1(Omega)notsubsetmathcal{C}^0(Omega)$.


3. If $d=3$ then $mathcal{H}^3(Omega)subsetmathcal{C}^0(Omega)$ but $mathcal{H}^2(Omega)notsubsetmathcal{C}^0(Omega)$.

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

Thanks in advance.

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