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Let $X$ be a ${\mathcal H}^m$-rectifiable subset of ${\bf R}^n$ and $x = \lim_i x_i$, where $x_i$ and $x$ are points possessing approximate tangent spaces.

Question Does it follow that $T_{x_i}X$ converges to $T_xX$? Or (in case it is not true in general): Under what assumptions does it hold?

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In general this can fail for all points: let $\{q_i\}$ be a countable dense set in the plane, and let $$X= \bigcup_i S(q_i,2^{-i}),$$ where $S(q,r)$ is the circle of center $q$ and radius $r$. Then $X$ is $1$-rectifiable, yet for all $x\in X$ and all $\delta>0$ you can find points in the ball $B(x,\delta)$ with tangent lines very different from $T_x X$ (in fact, you can find a dense set of tangent lines inside any such ball).

You can create a similar example with $X$ made up of a dense set of segments pointing in a dense set of directions inside any ball, and you can easily replicate this construction for any $n,m$.

Since stability under countable unions is a key feature of the notion of rectifiability, I doubt you can find a useful setting under which your statement holds.

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