It is true in one dimension that $H^1$ is continuously embedded in $L^{\infty}.$

Now, consider a compact interval $[0,1]$ with a partition $I_n:=([m/n,(m+1)/n])_{m \in \left\{0,...,n-1 \right\}}$ and a function $f$ on $I_n$ that is constant on every interval $[m/n,(m+1)/n].$

Then we can define the discrete derivative $f'(x)=n\left(\tfrac{f\left(\tfrac{x+1}{n}\right)-f\left(\tfrac{x-1}{n}\right)}{2}\right)$ and modify it accordingly on the boundary of the partition.

If we then define the $H^1_n$ norm on the space of such functions $$\Vert f \Vert _{H^1_n} := \sqrt{ \frac{1}{n} \sum_{i=1}^n \left(\left\Vert f\left( \frac{i}{n} \right) \right\Vert^2 +\left\Vert f'\left( \frac{i}{n} \right) \right\Vert^2\right) },$$

it is obvious that $H^1_n$ embeds of course continuously into $L^{\infty}.$

However, I am wondering whether $\Vert f \Vert_{L^{\infty}} \le C \Vert f \Vert_{H^1_n}$ holds for a constant independent of $n$ and $f$? This is to say, is this a discrete approximation of the Sobolev embedding theorem?