Study of geometric properties preserved under continuous deformations.
In topology, a function is continuous if you can draw its graph without lifting your pencil or making any jumps. This means small changes in input lead to small changes in output.
The formal definition: a function between two spaces is continuous if the pre-image of every open set is open. This sounds technical, but it just means the function doesn't "break" the space.
Continuity lets us stretch or squish shapes without tearing them apart, a key idea for comparing spaces in topology.
Stretching a rubber band into a different shape is a continuous transformation.
Mapping a circle onto a line by wrapping the circle around the line smoothly.
Continuous functions let us compare spaces without breaking them.