On sequences which are not uniformly converging on any open subset

We consider the property of nonuniform convergence to 0 of a sequence of functions on any open subset of a metric space. We consider three examples with respect to three different characteristics. Next we show that the three characteristics cannot be present simultaneously. For this purpose we introduce the so-called height function, which we use to quantify how far is a sequence of functions from satisfying any of the third characteristic. Moreover, we study properties of the height function and its relation to uniform convergence. Finally, we show that this quantification is precise.