1. The problem statement, all variables and given/known data Give an example of a non-continuous function on [0,1] that has no maximum and no minimum. 2. Relevant equations Well, a continuous function on a non-empty compact set will have a maximum and a minimum, so I guess this is why we need a non-continuous function. 3. The attempt at a solution Does this work? f(x) = 1/2 if x is a rational number x if x is an irrational number. Does this work? Breaking this up into 1/2s (if rational) and x's (if irrational) hopefully makes it non-continuous, and I suppose there is always some larger number that could be formed as we approach 1 (so no maximum). What do you think?