Q:

Does there exist a curve with non zero area?

Does there exist a curve with a bounded infinitely diffrentiable derivative (i.e. has a minimum  |curvature|) of hausdorf demension 2? Or even a diffrentiable curve of hausdorf demension 2?

A:

If $f: X \to Y$ is Lipschitz with Lipschitz constant $k$, i.e. $d(f(x),f(y)) \le k d(x,y)$, then for any $r$ we have $\mathcal H^r(f(X)) \le k^r \mathcal H^r(X)$, where $\mathcal H^r$ is $r$-dimensional Hausdorff measure.  Since any continuously differentiable function on an interval is locally Lipschitz, it follows that the image of a $C^1$ curve has $\sigma$-finite $1$-dimensional Hausdorff measure, and in particular Hausdorff dimension $1$.  
This can be generalized slightly: the image of any differentiable curve has $\sigma$-finite $1$-dimensional Hausdorff measure.  See this MO posting.

