Is there any other continuous function from R with standard topology to R with lower limit topology other than the constant function?
I can prove that no simple function (other than the constant one) is continuous.
But other functions I cannot prove. And I could not find it (if it exist).
[Math] continuous functions from R to R with different topologies
general-topology
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Best Answer
I think this is right. if f is continuous then for any a and b A=pre image of ([a,b[) is both open and closed. this means that A is empty or whole of R.this implies the existence of one such a and b such that A=R.which would then imply that f is constant otherwise there would exist c and d such that pre image of ([c,d[) is neither empty nor the whole of R.