Continuity and diameters of connected sets

The definition of uniform continuity (if it’s done right) can be phrased as: {f\colon X\to Y} is uniformly continuous if there exists a function {\omega\colon (0,\infty)\to (0,\infty)}, with {\omega(0+)=0}, such that {\textrm{diam}\, f(E)\le \omega (\textrm{diam}\, E)} for every set {E\subset X}. Indeed, when {E} is a two-point set {\{a,b\}} this is the same as {|f(a)-f(b)|\le \omega(|a-b|)}, the modulus of continuity. Allowing general sets {E} does not change anything, since the diameter is determined by two-point subsets.

Does it make a difference if we ask for {\textrm{diam}\, f(E)\le \omega (\textrm{diam}\, E)} only for connected sets {E}? For functions defined on the real line, or on an interval of the line, there is no difference: we can just consider the intervals {[a,b]} and obtain

{|f(a)-f(b)|\le \textrm{diam}\, f([a,b]) \le \omega(|a-b|)}

as before.

However, the situation does change for maps defined on a non-convex domain. Consider the principal branch of square root, {f(z)=\sqrt{z}}, defined on the slit plane {G=\mathbb C\setminus (-\infty, 0]}.

Conformal map of a slit domain is not uniformly continuous

This function is continuous on {G} but not uniformly continuous, since {f(-1 \pm i y) \to \pm i } as {y\to 0+}. Yet, it satisfies {\textrm{diam}\, f(E)\le \omega(\textrm{diam}\, E)} for connected subsets {E\subset G}, where one can take {\omega(\delta)=2\sqrt{\delta}}. I won’t do the estimates; let’s just note that although the points {-1 \pm i y} are close to each other, any connected subset of {G} containing both of them has diameter greater than 1.

These points are far apart with respect to the inner diameter metric

In a way, this is still uniform continuity, just with respect to a different metric. Given a metric space {(X,d)}, one can define inner diameter metric {\rho} on {X} by letting {\rho(a,b)} be the infimum of diameters of connected sets that contain both {a} and {b}. This is indeed a metric if the space {X} is reasonable enough (i.e., any two points are contained in some bounded connected set). On a convex subset of {\mathbb R^n}, the inner diameter metric coincides with the Euclidean metric {d_2}.

One might think that the equality {\rho=d_e} should imply that the domain is convex, but this is not so. Indeed, consider the union of three quadrants on a plane, say {A = \{(x,y) \colon x > 0\text{ or }y > 0\}}. Any two points of {A} can be connected by going up from whichever is lower, and then moving horizontally. The diameter of a right triangle is equal to its hypotenuse, which is the Euclidean distance between the points we started with.

A non-convex domain where inner diameter metric is the same as Euclidean

Inner diameter metric comes up (often implicitly) in complex analysis. By the Riemann mapping theorem, every simply connected domain {G\subset \mathbb C}, other than {\mathbb C} itself, admits a conformal map {f\colon G\to \mathbb D} onto the unit disk {D}. This map need not be uniformly continuous in the Euclidean metric (the slit plane is one example), but it is uniformly continuous with respect to the inner diameter metric on {G}.

Furthermore, by normalizing the situation in a natural way (say, {G \supset \mathbb D} and {f(0)=0}), one can obtain a uniform modulus of continuity for all conformal maps {f} onto the unit disk, whatever the domain is. This uniform modulus of continuity can be taken of the form {\omega(\delta) = C\sqrt{\delta}} for some universal constant {C}. Informally speaking, this means that a slit domain is the worst that can happen to the continuity of a conformal map. This fact isn’t often mentioned in complex analysis books. A proof can be found in the book Conformally Invariant Processes in the Plane by Gregory Lawler, Proposition 3.85. A more elementary proof, with a rougher estimate for the modulus of continuity, is on page 15 of lecture notes by Mario Bonk.

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