On Saturday, April 20, 2013 8:21:32 AM UTC+1, quasi wrote: > hbertaz wrote: > > > > >Doesn't anyone double-check these questions? What should be > > >done with the people whotook this exam and tried to answer an > > >unanswerable question like this one? > > > > It's answerable by an easy counterexample. > > > > In fact, the intent of the problem might simply have been to see > > if the student could recognize that a disproof was warranted, not > > a proof. > > > > Alternatively, it's possible that the actual problem included an > > additional hypothesis, unwittingly omitted, either by you or by > > whoever made the handout of sample test problems. > > >
Is the definition of "function" completely standardised or can the term "function" be legitimately defined as what others would call a "continuous function"?
In my opinion, the most likely thing is that the lecturer said and wrote [not on the exam paper, though] something like "All functions are assumed to be continuous."
I found the problem of proof for continuous functions moderately challenging. I was initially puzzled for about half an hour or so. Then it dawned on me to assume not globally 1-1, take x < y where f(x) = f(y), and use the fact that since the closed interval from x to y is compact, f attains its extrema there. Call one of the extrema x0 and act locally near that extremum to contradict locally 1-1. Fleshing out some more, use epsilon so that epsilon is much smaller than the distance between f(x0) and f(x).
The OP should realise that maths is not usually supposed to be a nitpicking exercise in translating everything into axioms. Maths is also a social language. If you've just done a course where all functions were continuous, then you would generally assume functions are continuous.
As quasi said, disproof by discontinuous counterexample, or continuous proof are both likely responses if you poll grad students. I suppose the optimal response is to do both of these things. In my opinion, if you only do one of them, proving the continuous case is a better response than just giving a discontinuous counterexample. After all, with some experience, it's clear that the question-setter meant that the function is continuous. Working out what people mean when they don't say exactly what they mean is part of what graduate-level mathematicians are supposed to learn.