(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that if f:X -> Y is a discontinuous bijection then f^{-1}:Y -> X is also discontinuous.

2. Relevant equations

N/A

3. The attempt at a solution

The contrapositive of this statement is that if f^{-1}:Y -> X is continuous then f:X -> Y is continuous. Since f is bijective its invertable; hence, all I need to prove is, if f:X -> Y is continuous then f^{-1}:Y -> X is continuous.

If f is continuous there exits a value delta for every epsilon such that delta = g*epsilon, where g is a function of epsilon.

We use the epislon delta method to prove f^{-1}is continuous. Hence, for all epsilon greater than zero there exists a delta greater than zero such that 0 < abs(y - k) < delta and 0 < abs(f^{-1}(y) - f^{-1}(k)) < epsilon. Hence, we may rewrite this as 0 < abs(f(x) - f(c)) < delta and 0 < abs(x - c) < epsilon. Since, this is merely the situation for f with epsilon and delta switched it follows that delta = epsilon/g. Therefore f^{-1}:Y -> X is continuous. Q.E.D.

I think some of my reasoning isn't particularly great, so any suggestions on how to correct this proof are welcome. Thanks.

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# Inverse proof

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