Editing Continuous function (section)

===Properties===
If <math>f : X \to Y</math> and <math>g : Y \to Z</math> are continuous, then so is the composition <math>g \circ f : X \to Z.</math> If <math>f : X \to Y</math> is continuous and
* ''X'' is [[Compact space|compact]], then ''f''(''X'') is compact.
* ''X'' is [[Connected space|connected]], then ''f''(''X'') is connected.
* ''X'' is [[path-connected]], then ''f''(''X'') is path-connected.
* ''X'' is [[Lindelöf space|Lindelöf]], then ''f''(''X'') is Lindelöf.
* ''X'' is [[separable space|separable]], then ''f''(''X'') is separable.

The possible topologies on a fixed set ''X'' are [[partial ordering|partially ordered]]: a topology <math>\tau_1</math> is said to be [[Comparison of topologies|coarser]] than another topology <math>\tau_2</math> (notation: <math>\tau_1 \subseteq \tau_2</math>) if every open subset with respect to <math>\tau_1</math> is also open with respect to <math>\tau_2.</math> Then, the [[identity function|identity map]]
<math display="block">\operatorname{id}_X : \left(X, \tau_2\right) \to \left(X, \tau_1\right)</math>
is continuous if and only if <math>\tau_1 \subseteq \tau_2</math> (see also [[comparison of topologies]]). More generally, a continuous function
<math display="block">\left(X, \tau_X\right) \to \left(Y, \tau_Y\right)</math>
stays continuous if the topology <math>\tau_Y</math> is replaced by a [[Comparison of topologies|coarser topology]] and/or <math>\tau_X</math> is replaced by a [[Comparison of topologies|finer topology]].