3 · Check continuity with a sequence

The sequence points approach a limit. A continuous map must send those points toward the value assigned to the limit point.

Try this. Choose where the sequence images approach, then choose the image of the limit. Try unequal values first. Make them equal and see why the continuity result changes.

Remark 1.6 says that convergent sequences determine the topology of every metrizable space. For spaces defined by a distance, this activity shows the full sequence test for continuity.