Let g be defined on a set containing the range of a function f. If f is continuous at z0 and g is continuous at f(z0), then g(f(z)) is continuous at z0.

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- February 14th 2010, 08:40 PMjzelltFunction proof
Let g be defined on a set containing the range of a function f. If f is continuous at z0 and g is continuous at f(z0), then g(f(z)) is continuous at z0.

- February 14th 2010, 08:59 PMJhevon
- February 15th 2010, 12:15 PMjzellt
The definition of continuity we're unsing involves limits and is NOT the epsilon delta method.

I havn' really gotten anywhere because I havn't a clue how to do this.. - February 15th 2010, 02:05 PMJoachimAgrell
Does it involve sequences? I mean (given

*f:X->*,**R***a*in the set of limit points of*X*), did you define limits like :

if, and only if, for all sequence such that and we have ? - February 15th 2010, 06:42 PMmabruka
Maybe his definition of continuity is

"f is continuous in a iff " - February 15th 2010, 06:53 PMDrexel28
This is a classic and easy to find on the internet theorem...but let met give you an outline of the proof.

Let and let where are both continuous. (we actually only have to have where but that is neither here nor there). Then is also continuous.

Proof: Let be arbitrary. Since and is continuous there exists some such that . And since is continuous there exists some such that and so

Or much nicer, if you are using the more topological definition (which is equivalent to the regular in metric spaces) that is continuous iff is open in whenever is open in merely note that and is ope in and so is open in . Done.