there is no such thing as a linear relationship

Linear relationships only exist as simplifications of non-linear relationships or simplifications of portions of non-linear relationships. Even our own representations of lines themselves are in fact non-linear. however, thinking of things in terms of lines is helpful. Apparently it's useful to mathematicians, so lets leave them with their lines. But in general we come into problems when we try to apply linear models to anything in reality because nothing is linear. One possibility for a truly completely strait line is the edge of a crystal. but who's to say that the atoms line up exactly, and further, the edges of an atom are not definite but flexible, a line is not flexible, nor rigid, it is imaginary and only exists as a tool that can be used properly or improperly in explaining or simplifying the universe.

why should you care and what new information is this:
Well, in economics and in any quantifiable study graphs are usually employed to represent the data visually. Students are taught from a young age the idea of lines and that definition is regularly redefined for them. In the context of graphs, lines are no longer used but curves are substituted. Unfortunately, similar definitions of lines remain on curves, such as zero width. Because of this simplistic model of reality we have to employ cumbersome and outmoded tools such as error bars. Whenever a simplification is to take place, it is important that the students know that such simplification is taking place and that they see the limits of the lens you've given them rather than believing the image they currently see is the be all and end all description of reality.