According to the definition of continuity in topology a function f:X -> Y is continous if for every open set V in Y, the set ##f(V)^{-1}## is open in X. I have found this definition powerfull in dealing with proofs of a more general nature, but when presented with the trivially discontinous function from the reals to the reals(adsbygoogle = window.adsbygoogle || []).push({});

$$f(x) = 1 \ \text{for} \ x< 0, \ 0 \ \text{for} \ \ x \geq 0 $$

I have problems to argue why this function is obviously not continuous from the above definition.

Which open set do I choose in it's codomain which is not open in it's domain?

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# Establishing that a function is discontinous

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