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.