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Reading both this answer and this answer to an old question, I wonder:

Is the length of curves in the Manhattan metric a continuous function of the curve?

Note that I don't place any requirements on the curves, apart from the obvious one that they need to have a well-defined length under the Manhattan metric.

Edit:

I had assumed that there's a natural topology for curves. But since 6005 asked for the curve topology in the comments, I gather that this is not the case. Therefore I'll specify the topology as follows:

I assume all curves to be continuous functions from closed intervals (so maxima and minima always exist).

I define the distance of a point to a curve as the minimum of the Manhattan distances of that point to the points of the curve.

I define the distance of two curves as the maximum of the distance of the points of one of the curves to the other curve.

I define the topology of the set of curves as the topology induced by the distance between curves I just defined.

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Even using the Manhattan metric to measure spatial distance, you ultimately run into the same problem: modifying a curve by adding small but rapid oscillations adds quite a lot to the path length without moving the curve very "far" in the metric on curves you've defined.

To make sense of what's going on, you need to realize that the choice of metric on the curves is what really matters. Let $d$ be your metric in $\mathbb{R}^2$, whether it's Euclidean, Manhattan, or something else. We want to look at different possible metrics on the set of continuous functions $[0,1] \to \mathbb{R}^2$.

You described a metric in your edit. In mathematical notation, that metric was: $$\rho(f,g) = \sup_{x \in [0,1]} d(f(x), g(x))$$ That is, the distance between $f$ and $g$ is the furthest those functions ever get from one another.

The problem is that the above metric only "sees" when the functions take different values; it can't "see" if one is darting around more quickly than the other and accumulating a lot more path length. To fix that, we can build another metric, one that sees differences in both "location" and "speed." (For simplicity, let's assume we only care about continuously differentiable functions.) $$\rho(f,g) = \sup_{x\in[0,1]} d(f(x),g(x)) + \sup_{x \in (0,1)} \lVert f'(x) - g'(x)\rVert$$ ($\lVert \cdot \rVert$ denotes a the norm of a vector. Recall that since $f,g$ are functions into $\mathbb{R}^2$, their derivatives live in $\mathbb{R}^2$, also. Note that you have a choice of which norm to use akin to the choice of Euclidean versus Manhattan metric.) With this metric on paths, the path length is continuous: note in particular that adding a rapid but small oscillation requires changing $f'$ considerably.

Studying different metrics on sets of functions is just the tip of a wonderful iceberg called functional analysis.

  • Hmmm … if I just added $|l(f) - l(g)|$ to the metric, it also should see the difference, without requiring differentiability. However that has the smell of cheating ;-) – celtschk Sep 19 '16 at 21:48
  • @celtschk There are ways to drop the differentiability requirement without such "cheating." For example, you might work with Lipschitz continuous functions instead. If you have some experience with real analysis, try and figure out how to modify the metric I gave to work with Lipschitz functions instead of continuously differentiable ones. –  Sep 20 '16 at 06:46