I need to do some euclidean distance on a global scale. I find that there is no a natural continuation at the edges due to the Maximum distance option set to default. I've tried with different values always with the same output.
I wonder if someone can suggest a way round this so I can get a continuous raster dataset instead of a sharp differentiation at the edges.
The Euclidean Distance tool isn't geodesic aware, and assumes that the analysis is being performed Cartesian space (i.e. there's no relationship between -180 and 180). There are a few ways to fix this, perhaps the simplest is to use the tools within Data Management > Projections and Transformations > Raster, in particular Shift, to create a version which spans 0-360 degrees, centered on the -180/180 anti-meridian. From there, you should be able to perform a second Euclidean distance calculation and get more sensible results, then merge those two results back together.
But probably better than just doing this is to use a projected space for your analysis -- in unprojected space, measurements of distance vary both by latitude and longitude, so particularly as you head to the poles, the distance of one cell shifts pretty dramatically. A couple of ways of dealing with this include rescaling distances by how far they are from the equator, or using a projected space for your calculations. Using a projection is probably your best bet if you need precision in the answer.