<p>The purpose of this paper is to extend the definition of quasi-arithmetic means by taking a strictly monotone generating function instead of a strictly monotone and continuous one. We establish the properties of such means and compare them to the analogous properties of standard quasi-arithmetic means. The comparability and equality problems of generalized quasi-arithmetic are also solved. We also provide an example of a mean which, depending on the underlying interval or on the number of variables, could be or could not be represented as a generalized quasi-arithmetic mean.</p>