Counterexample for Hausdorff Assumption is removed

Question

I had the following fact:

$A\subset X,f:A\to Y$ , where Y is Hausdorff space .f is continuous function on A. If there is continuous extension of f as $g:\bar A\to Y$ exist .Then this extension is unique.

I know that such extension need not exist .

I was thinking all assumption kept same and if we relaxed Hausdorff condition then the result must not hold.

I tried to find the counterexample.

I tried finite point non Hausdorff topology, But still not successful.I would be thankful if someone helps me to find such an example.

Thanks a lot

Answer 1

If $Y$ is the real line (or any set with more than one point) with indiscrete topology then any function into $Y$ is continuous. Obviously the extension is not unique in this case.