Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math.independent

Topic: Slightly OT: Double-Check on Qual Exams.
Replies: 14   Last Post: Apr 22, 2013 3:35 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Paul

Posts: 401
Registered: 7/12/10
Re: Slightly OT: Double-Check on Qual Exams.
Posted: Apr 22, 2013 12:28 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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.

Paul Epstein







Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.